LINEiH 

PERSPECTIVE 


Digitized  by  the  Internet  Archive 


in  2015 

https://archive.org/details/linearperspectivOObart 


LINEAR  PERSPECTIVE 

EXPLAINED. 


BY 

WM.  N.  BARTHOLOMEW, 

AUTHOR   OF   BARTHOLOMEW'S   SKETCH   BOOK,  AND   SERIES   OF   DRAWING  BOOKS, 

IN    SIX  NUMBERS. 


BOSTON: 

CYRUS    G^.  OOOKE. 

18  6  6. 


Entered,  according  to  Act  of  Congress,  in  the  year  1859,  by 
William  N.  Bartholomew, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts, 


ELECTROTYPED  AT  THE 
BOSTON    STEREOTYPE  FOUNDRY. 


MaETTYCENre 


PREFACE 


It  is  a  matter  well  known  to  all  who  have  given  attention  to 
the  subject^  that  a  very  general  and  strongly-marked  increase  of 
interest  has  been  manifested  in  the  study  of  Drawing  within  a 
few  years  past  in  the  United  States.  The  time  has  come  when 
some  practical  knowledge  of  drawing  is  getting  to  be  generally 
recognized  by  the  public  as  an  essential  part  of  every  thorough 
system  of  education. 

Drawing  is  now  very  generally  taught  both  in  our  public  and 
private  schools.  There  is^  however,  no  branch  of  the  art  more 
worthy  of  attention  than  that  which  forms  the  subject  of  this 
work,  and  there  is  none  which  receives  so  little.  The  ambi- 
tion to  have  something  to  shoiv^  or  the  desire  to  make  something 
fretty^  misleads  multitudes  from  a  proper  and  systematic  course 
of  instruction.  Those  fundamental  principles  which  lie  at  the 
basis  of  the  art,  and  which  teach  us  how  we  may  accurately 
represent  the  forms  and  proportions  of  objects,  are  entirely 
overlooked.  Weeks,  and  oftentimes  months,  are  spent  in  copying, 
line  for  line,  and  dot  for  dot,  some  old  ruin,  and  its  usual  accom- 
paniments. The  whole  field  of  Art,  save  that  which  relates  to 
color,  is  run  over  without  the  knowledge  of  a  single  principle 
involved.  Those  who  have  attempted  the  study,  have,  in  most 
instances,  spent  their  time  in  learning  to  copy  the  characters  used 


4 


PREFACE. 


in  the  expression  of  ideas,  instead  of  studying  the  principles  of 
the  art.  They  have  spent  their  time  in  pursuing  the  shadow,  to 
the  neglect  of  the  substance. 

It  has  been  the  aim  of  the  author,  in  preparing  this  work,  to 
furnish  the  young  with  a  text-book  designed  to  impart  a  prac- 
tical knowledge  of  the  art  of  making  truthful  pictures  of  objects. 
We  have  aimed  to  place  the  subject  within  the  reach  of  the  boy 
of  twelve  years,  and^  in  doing  this,  have  studied  to  avoid  the  error 
into  which  many  have  fallen  in  their  efforts  to  make  the  subject 
easy  of  comprehension  —  that  of  superficiality. 

This  work  differs  from  all  others  with  which  we  are  acquainted 
in  one  or  more  of  the  following  particulars :  — 

It  contains  a  full  explanation  of  first  principles. 

No  principle  is  used  in  the  explanation  of  another,  which  has 
not  itself  been  explained. 

The  problems  given  are  of  a  practical  character. 

The  method  of  sketching  objects  is  explained  in  connection 
with  the  method  of  determining  their  perspectives  by  means  of 
vanishing  points. 

In  determining  the  perspectives  of  objects,  a  reason  is  given 
why  every  line  is  drawn  as  it  is. 

All  explanations  are  as  concise  as  possible,  and  in  language 
that  all  may  understand. 

At  the  close  of  the  book  may  be  found  a  series  of  questions. 

Boston,  July  27,  1859. 


t 


CONTENTS. 

t 


INTRODUCTION. 

PAGE 

Drawing  Instruments,  their  Use  and  Management   7 

Paper.  —  Lead  Pencil.  —  Rubber.  —  Rule.  —  Right-angled  Triangle.  —  Compasses. 

LINEAR  PERSPECTIVE. 

CHAPTER  I. 

Definition  of  Terms   11 

Linear  Perspective.  —  Plane.  —  Ground  Plane.  —  Picture  Plane.  —  Point  of  View.  — 
Centre  of  View.  —  Horizon.  —  Distance  of  Picture.  —  Point  of  Distance.  —  Visual 
Rays.  —  Vanishing  Point.  —  Elevation,  —  Plan. 

CHAPTER  II. 

Principles  of  Linear  Perspective  developed   14 

Optics  the  basis  of  perspective.  —  Phenomena  of  vision  explained.  —  Method  of  deter- 
mining the  perspectives  of  points  by  means  of  visual  rays.  —  On  finding  the  perspectives 

of  Hnes,  straight  or  curved  The  perspective  of  a  square  situated  parallel  to  the  picture 

plane  determined.  —  The  perspective  of  the  square  examined,  and  rules  framed,  based 
on  principles  developed.  —  The  perspective  of  a  square  situated  oblique  to  the  picture 
plane  determined.  —  The  perspective  of  the  square  examined,  and  rules  framed,  based  on 

principles  developed  The  perspectives  of  the  squares  examined,  and  a  rule  framed 

based  on  a  principle  developed. 

CHAPTER  III. 

Preparation  of  the  Paper  for  the  Representation  of  Objects  by  the  Use  of 

Vanishing  Points   22 

On  drawing  the  base  line.  —  On  placing  the  centre  of  view,  and  drawing  the  horizon  

On  drawing  plans  and  elevations.  —  On  placing  the  point  of  distance.  —  Lines  and 
points  necessary  in  working  a  drawing. 

(5) 


6 


CONTENTS. 


CHAPTER  IV. 

The  Proper  Place  in  the  Picture  for  the  Centre  of  View,  and  the  Proper 

Distance  to  assume  as  the  Distance  of  Picture   24 

The  location  of  the  actual  centre  of  view  as  pictured  on  the  retina  of  the  eye,  considered.  — 
Rule  for  placing  the  centre  of  view.  —  Remarks  on  the  application  of  the  rule.  —  The 
distance  at  which  an  object  must  be  from  the  eye,  to  be  seen  distinctly,  considered.  Rule 
for  determining  the  proper  distance  to  assume  as  the  distance  of  picture. 

CHAPTER  V. 

On  the  Method  of  de^rmining  the  Perspectives  of  Points  25 

The  principle  on  which  the  perspectives  of  points  are  found. 

PROBLEMS. 


PARALLEL  PERSPECTIVE. 


pROB.  1.   It  is  required  to  find  the  perspective  of  a  point  on  the  ground  plane   26 

Prob.  2.    It  is  required  to  find  the  perspective  of  a  square   27 

Prob.  3.    It  is  required  to  find  the  perspective  of  a  rectangle   27 

Prob.  4.   It  is  required  to  find  the  perspectiveof  a  point  above  the  ground  plane.  —  Method 
of  determining  the  perspectives  of  vertical  lines.  28 


Prob.  5.  It  is  required  to  find  the  perspective  of  a  cube.  — The  perspective  of  the  cube  ex- 
amined, and  rules  framed,  based  on  principles  developed.  —  Drawing  from  objects  recom- 
mended. —  Method  of  sketching  the  cube.  —  On  determining  the  accuracy  of  a  drawing. 

—  Suitable  subjects  for  sketching.  —  The  proper  course,  to  pursue  in  drawing  many  ob- 
jects illustrated  by  the  drawing  of  a  chest.  —  Drawing  from  copies  recommended   29 

Prob.  6.    It  is  required  to  find  the  perspective  of  a  cube  and  the  perspective  of  a  pyramid. 

—  The  method  of  sketching  a  pyramidical  roof,  or  any  object  in  the  form  of  the  pyra- 


mid  34 

Prob.  7.    It  is  required  to  find  the  perspective  of  a  building   36 

Prob.  8.    It  is  required  to  find  the  perspective  of  a  building,  with  a  door,  windows,  and 

chimney,  the  front  wall  of  the  building  being  parallel  to  the  picture  plane   38 

Prob.  9.    It  is  required  to  find  the  perspective  of  the  building  referred  to  in  Prob.  8,  the 

end  wall  being  in  the  picture  plane.  —  Instructions  relating  to  the  sketching  of  buildings.  41 


OBLIQUE  PERSPECTIVE. 

Prob.  10.  It  is  required  to  find  the  perspective  of  a  cube.  —  Method  of  sketching  the  cube. 
—  Method  of  sketching  various  objects  illustrated  by  the  drawing  of  a  chair.  —  An  exer- 
cise in  drawing  from  the  copy  <   44 

Prob.  11.  It  is  required  to  find  the  perspective  of  a  building.  —  Method  of  sketching 
buildings   48 


OBJECTS  CONTAINING  CURVED  LINES. 

Prob.  12.  It  is  required  to  find  the  perspective  of  a  circle.  — Method  of  determining  the 
perspective  of  an  octagon.  —  The  perspective  of  the  circle  examined.  —  Method  of  sketch- 
ing the  circle   54 

Prob.  13.  It  is  required  to  find  the  perspective  of  a  cylinder.  —  Method  of  sketching  cylin- 
drical objects  i . .  i  »  56 


f 


INTRODUCTION. 


DRAWING  instruments!— THEIR  USB  AND  MANAGEMENT. 

1.  Paper.  Paper  should  have  just  tooth  enough  to  admit  of 
making  a  clear,  bright  line.  There  is  a  paper  known  as  car- 
tridge  paper y  which,  for  ordinary  purposes,  will  meet  the  wants 
of  the  pupil;  for  sketching,  there  is  nothing  better  than  the 
German  cartoon. 

In  drawing,  the  paper  should  be  placed  upon  a  flat,  hard 
surface,  for  the  following  reasons:  A  straight  line  cannot  be 
drawn  with  certainty  when  the  paper  is  placed  upon  a  yielding 
surface.  It  is  impossible  to  draw  an  even  line  when  the  paper 
is  so  placed,  and  the  paper  is  sure  to  break  or  dent  under  the 
touch  of  the  pencil. 

Paper  should  never  be  rolled  if  it  can  possibly  be  avoided,  as 
it  always  cockles  on  being  unrolled. 

2.  Lead  Pencil.  The  selection  of  a  pencil  is  a  matter  of  im- 
portance. A  pencil  should  be  capable  of  making  a  clear,  jetty 
line;  it  should  be  tough,  and  free  from  grit.  A.  W.  Faber's 
pencils  excel  all  others  in  these  desirable  qualities,    The  dif- 

(7) 


8 


INTRODUCTION. 


ferent  degrees  of  hardness  are  indicated  either  by  letters  or 
numerals.  Those  marked  H  H  H,  and  those  marked  4,  are 
admirably  adapted  for  making  fine  and  dehcate  lines.  For 
sketching,  the  H  or  3  pencil  is  all  that  is  desired. 

In  drawing  diagrams,  the  pencil  should  have  a  fine,  delicate 
point,  and  the  wood  of  the  pencil  should  be  scarfed  back  at 
least  three  fourths  of  an  inch  from  the  point. 

In  sharpening  the  pencil,  let  it  rest  firmly  against  the  ball  of 
the  thumb  of  the  right  hand,  in  such  a  manner  as  to  support 
the  lead  of  the  pencil.  ^ 

3.  Rubber.  Eubber  is  the  ordinary  instrument  for  cleaning  a 
drawing,  and  for  erasing  marks  of  the  pencil.  To  meet  the 
wants  of  the  draughtsman,  the  rubber  should  readily  remove 
particles  of  lead  from  the  paper;  and  it  should  wear  away  as 
used,  so  as  to  leave  the  rubber  perfectly  clean  after  use.  The  best 
to  be  had  is  that  manufactured  by  the  Union  Rubber  Company. 

Rubber  should  never  be  used  if  it  can  possibly  be  avoided,  as 
the  best  of  rubber  materially  injures  the  surface  of  the  paper. 
After  it  has  been  once  employed  to  any  extent,  it  is  almost 
impossible  to  keep  the  paper  clean.  To  avoid,  as  much  as  may 
be,  the  necessity  of  resorting  to  its  use,  no  more  lines  should  be 
made,  in  working  the  drawing,  than  are  absolutely  required,  and 
all  lines  not  forming  a  part  of  the  outline  of  the  drawing  should 
be  made  as  light  as  is  consistent  with  the  distinctness  of  the  work. 

4.  Rule.  The  pupil  will  find  it  convenient  to  have  two  rules; 
one  should  be  at  least  two  feet  long,  and  the  other  from  six  to 
eight  inches  in  length.    In  drawing  long  lines,  the  former  is 


INTRODUCTION. 


9 


indispensable;  but  for  common  use^,  the  latter  will  be  found  far 
more  convenient. 

In  selecting  a  rule,  great  care  should  be  taken  to  obtain  one 
perfectly  straight,  inasmuch  as  the  accuracy  of  the  drawing 
depends  greatly  on  the  straightness  of  the  lines. 

5.  Eight-angled  Triangle.  This  instrument  is  used  in  drawing 
perpendiculars  and  parallels.  The  size  most  convenient  measures 
from  four  to  six  inches  on  a  side. 

The  value  of  this  instrument  depends  upon  the  accuracy  of 
the  right  angle,  and  the  straightness  of  its  sides.  To  test  the 
accuracy  of  ^  the  right  angle,  place  one  side  of  the  right  angle 
against  the  edge  of  the  rule,  and  draw  a  fine,  delicate  line  along 
the  edge  of  the  other  side  of  the  right  angle;  then,  holding  the 
rule  firmly  down,  turn  the  other  side  of  the  triangle  up,  and  bring 
the  same  edge  up  to  either  extremity  of  the  line,  and  draw  a 
light  line,  as  before;  if  the  two  lines  thus  drawn  form  one  line 
of  the  same  width  throughout,  the  instrument  is  perfect ;  if  other- 
wise, it  is  imperfect  and  worthless. 

The  triangle  is  used  thus:  Suppose  it  is  required  to  draw  a 
perpendicular  from  a  given  point  in  a  straight  line.  First,  place 
the  edge  of  the  rule  on  the  straight  line;  then  place  one  side 
of  the  right  angle  of  the  triangle  up  against  the  edge  of  the 
rule,  and  sUde  the  triangle  along  to  the  given  point,  and  draw 
the  line. 

Suppose  it  is  required  to  draw  a  line  parallel  to  a  given  line. 
Place  the  triangle  against  the  edge  of  the  rule,  as  before,  and 
bring  either  one  of  the  remaining  edges  of  the  triangle  to 
coincide  with  the  given  line ;  then,  holding  the  rule  firmly  down, 
2 


10 


IN^TRODUCTION. 


slide  the  triangle  along  to  the  point  from  which  the  line  is  to 
be  drawn,  and  draw  the  line. 

6.  CoMPASSjES.  The  pupil  should  be  provided  with  a  pair  of 
compasses  so  made  that  a  pencil  may  be  attached  to  one  of  its 
legs.  It  is  convenient  to  have  two  pairs  of  compasses  —  one 
small,  and  the  other  large;  but  if  only  one  pair  be  used,  they 
should  be  of  such  a  size  that,  when  opened  as  far  as  practi- 
cable, the  distance  from  point  to  point  will  be  five  or  six  inches. 

This  instrument  is  used  in  measuring  and  transferring  dis- 
tances, and  in  describing  circles.  The  compasses  should  be  held 
gently  at  the  joint,  between  the  thumb  and  forefinger,  in  such 
a  manner  as  not  to  press  in  the  least  against  the  legs.  When 
held  in  this  way,  the  distance  between  the  points  will  not  be 
altered. 


LINEAR  PERSPECTIVE. 


CHAPTER  I. 

DEFINITION  OF  TERMS. 

7.  Linear  Perspective  is  an  art  wliicli  relates  to  the  representation  of 
the  outlines  of  objects,  in  such  a  manner  that  the  representation,  when  seen 
from  a  particular  point,  shall  present  to  the  eye  the  same  appearance  as  that 
presented  by  the  objects  themselves.  The  representation  of  an  object  so 
made  is  called  its  perspective^  and  any  line  or  point  in  the  drawing,  repre- 
senting a  line  or  point  in  the  object,  is  called  the  perspective  of  the  line  or 
point.  When  the  perspective  of  a  line  is  prolonged  indefinitely,  it  is  called 
the  indefinite  perspective  of  the  line. 

8.  A  Plane  is  either  a  real  or  an  imaginary  surface,  in  which,  if  two 
points  be  assumed  at  pleasure,  and  connected  by  a  straight  line,  that  line  will 
lie  wholly  in  the  surface.  Every  plane  may  be  supposed  to  be  produced  in 
any  or  in  every  direction,  as  convenience  requires. 


Fig.  1. 


9.    The  Ground  Plane  is  either  a  real  or  an  imaginary  plane,  upon 

which  the  objects  to  be  represented  rest.  Its  position  is  hopizontal.  In 

Fis:.  1,  N  IJ  K  represents  this  plane. 
^  (11) 


12 


DEFINITION  OF  TERMS. 


10.  The  Picture  Plane  is  an  imaginary,  transparent  plane,  on  which 
the  objects  are  delineated.  It  is  supposed  to  stand  perpendicular  to  the 
ground  plane,  fronting  the  observer,  and  between  him  and  the  objects  to  be 
represented.  In  Fig.  1,  A  B  L  D  represents  this  plane.  The  line  B  L, 
formed  by  the  intersection  of  this  plane  with  the  ground  plane,  is  called 
the  base  line  of  the  picture  plane. 

11.  The  Point  of  Yiew  is  the  point  from  which  the  eye  is  supposed  to 
view  the  objects  to  be  placed  in  perspective.  In  Fig.  1,  the  eye  of  the 
observer  represents  this  point. 

12.  The  Centre  of  View  is  a  point  to  which  the  eye  of  the  observer  is 
supposed  to  be  directed,  when  looking  in  a  direction  perpendicular  to  the 
picture  plane.    In  Fig.  1,  the  point  C  represents  this  point. 

13.  The  Horizon.  If  one  were  so  placed  as  to  command  an  unob- 
structed view  of  a  horizontal  plane,  stretching  away  as  far  as  the  eye  could 
see,  the  water  or  ground  before  him  would  appear  to  incline  upwards  from 
the  spot  on  which  he  stood,  and  the  limit  of  the  plane,  in  the  extreme  dis- 
tance, would  be  bounded  by  an  apparently  straight,  horizontal  line,  on  a  level 
with  the  eye,  called  the  horizon.  The  horizon  always  appears  to  be  on  a 
level  with  the  eye,  whether  the  view  be  had  from  the  low  land  or  the  moun- 
tain top.  The  term  horizon  is  also  applied  to  the  perspective  of  this  line. 
In  Fig.  1,  the  line  H  R  represents  this  line. 

14.  The  Distance  of  Picture  is  the  supposed  distance  of  the  picture 
plane  from  the  observer.  In  Fig.  1,  the  dotted  line  drawn  from  the  eye  to 
C,  measures  the  distance  of  picture. 

15.  The  Point  of  Distance  is  a  point  taken  in  the  perspective  of  the 
horizon,  on  either  side  of  the  point  representing  the  centre  of  view,  and  at 
a  distance  from  it  equal  to  the  supposed  distance  of  the  picture  plane  from 
the  observer. 

16.  Visual  Rays  are  those  rays  of  light  which  pass  from  the  object  to 
the  eye  of  the  observer.    Visual  rays  are  represented  by  straight  lines. 

17.  A  Vanishing  Point  is  any  point  on  the  picture  plane  where  the 
perspectives  of  parallel  lines  meet  on  being  prolonged. 

18.  An^  Elevation  is  a  drawing  giving  the  actual  form,  and  the  exact 


DEFINITION  OF  TERMS. 


13 


proportions,  of  the  different  upright  parts  of  any  upright  side  of  a  building 
or  other  object. 


Pig.  2. 


Fig.  3. 


In  Fig,  2,  we  have  a  front  elevation  of  the  building  represented  iii 
Fig.  3 ;  and  in  Fig.  4,  we  have  a  side  elevation  of  the  building. 

19.    A  Plan  is  a  drawing  constructed  from  a  horizontal  section,  or 
from  horizontal  sections,  of  a  building  or  other  object. 
For  the  purpose  of  perspective  drawing,  the  plan  of  the  object  to  be  rep- 


 -I-  - 


E 


Pig.  4. 


Pig.  5. 


resented  must  indicate  every  change  of  form  that  may  occur  by  cutting  the 
object  horizontally  at  any  point. 

In  Fig.  5,  we  have  a  plan  of  the  building  represented  in  Fig.  3.  This 
plan  was  made  from  the  figures  formed  by  cutting  the  building  at  two  points 
—  one  of  these  points  being  just  above  its  base,  passing  through  the  door 
and  windows,  as  indicated  by  the  dotted  lines  D  E  and  E  F,  and  the  other 
through  the  line  of  the  ridge  H  I.    The  double-lined,  rectangular  figure 


I.INEAR  PERSPECTIVE. 


A  B  C  D,  Fig-,  5,  is  a  plan  of  the  walls  of  the  building;  the  open  space  E, 
of  the  doorway ;  the  single  lines  F  and  H,  of  the  windows ;  the  figure 
niarked  I,  of  the  chimney ;  and  the  line  J  K,  of  the  ridge. 

From  the  three  geometrical  drawings,  the  front  elevation,  the  side  elevation,  and  the  plan,  the 
proportions  of  the  building  may  be  known. 

The  length  of  the  building  is  shown  in  the  plan,  and  in  the  front  elevation. 

The  breadth  of  the  building  is  shown  in  the  plan,  and  in  the  side  elevation. 

The  height  of  the  walls  is  shown  in  the  front  and  side  elevations. 

The  vertical  height  of  the  roof  is  shown  in  the  front  and  side  elevations. 

The  pitch  of  the  roof  is  shown  in  the  side  elevation. 

The  height  of  the  chimney  is  shown  in  the  front  and  side  elevations. 

The  width  of  the  front,  and  back  face  of  the  chimney,  is  shown  in  the  plan,  and  in  the  front 
elevation. 

The  width  of  the  faces  of  the  chimney  parallel  to  the  end  walls  of  the  house  is  shown  in  the 
plan,  and  in  the  side  elevation. 

The  width  of  the  windows,  the  width  of  the  door,  and  their  situation,  is  shown  in  the  plan, 
and  in  the  front  elevation. 

The  height  of  the  door,  the  height  of  the  windows,  and  their  height  above  the  base  of  the 
building,  is  shown  in  the  front  elevation. 


CHAPTER  II, 
PRINCIPLES  OF  LINEAR  PERSPECTIVE  DEVELOPED. 

20.  The  principles  which  underlie  and  form  the  basis  of  linear  perspec- 
tive are  few  in  number,  and  easy  of  comprehension.  They  are  deduced 
from  the  science  of  optics,  and  depend  upon  the  following  truths :  

When  an  object  is  exposed  to  the  light,  rays  are  reflected  from  each  point 
in  the  surface,  in  every  possible  direction. 

Rays  of  light,  while  passing  through  the  same  medium,  proceed  in  straight 
lines. 

When  the  eye  is  directed  towards  an  obr 
ject,  rays  from  every  visible  point  enter  the 
eye,  and  form  upon  the  retina  an  image  or 
picture  of  the  object,  and  the  impression 
^  thus  made  produces  the  sensation  of  sight. 

21.  To  illustrate  these  truths,  let  the 
circle  in  Fig.  6,  represent  a  vertical  section 
of  the  eye,  the  point  E,  its  pupil,  and  let 
A  B  be  a  straight  line,  towards  which  the  eye  is  directed. 


A 


B 

Pig.  6. 


• 


PRINCIPLES  DEVELOPED. 


15 


From  the  point  A  we  may  suppose  rays  of  light  to  ho  reflected  in  every 
direction  ;  a  part  of  these  rays  enter  the  eye,  through  the  pupil,  one  of 
which  takes  the  direction  of  the  line  A  D,  and  there  is  formed  upon  the 
retina  at  D,  the  image  of  the  point  A.  Prom  the  point  B,  rays  enter  the 
eye,  and  there  is  formed  upon  the  retina  at  C,  the  image  of  tlie  i)oint  15. 
In  Hke  manner,  rays  from  each  point  in  the  line  A  B  enter  the  eye,  and  lall 
upon  the  retina  hetween  the  points  C  and  D,  completing  the  image  C  D. 

22.  Let  us  now  suppose  one  viewing  an  object  through  a  transparent 
plane,  placed  upright  before  him.  From  every  visible  point  in  the  object  a 
ray  of  light  proceeds  in  a  right  line  to  the  eye,  passing  through  a  point  in 
the  plane.  If  to  each  point  in  the  plane,  pierced  by  the  visual  rays,  liglit 
and  shade  could  be  given,  similar  to  that  which  characterizes  the  points 
from  which  the  rays  proceed,  the  rays  from  the  picture  thus  produced  would 
form  in  the  eye  an  image  precisely  like  that  formed  by  the  object  itself. 
The  picture,  therefore,  would  be  a  perspective  drawing  of  the  object. 

This  truth  may  be  illustrat- 
ed. Let  the  circle  in  Fig.  7  rep- 
resent the  eye,  A  C  a  straight 
line,  towards  which  the  eye 
is  directed,  and  let  the  figure 
I  J  K  L  represent  a  transparent 
plane. 

As  already  shown,  the  visual 
rays  from  the  line  A  C  form  on 
the  retina  the  image  F  H.  The 
visual  ray  from  the  point  A,  in 
its  passage  to  the  eye,  passes  through  the  plane  at  the  point  a,  and  the 
visual  ray  from  C  passes  through  the  plane  at  c.  Since  A  C  is  a  straight 
line,  the  visual  rays  from  the  points  between  A  and  C  pass  through  the 
plane  at  points  between  a  and  c ;  therefor^,  if  the  points  a  and  c  be  con- 
nected by  a  straight  line,  the  line  a  c  includes  all  points  in  the  plane 
pierced  by  the  visual  rays.  Now,  suppose  the  line  A  C  removed ;  the  vis- 
ual rays  from  the  line  a  c  form  the  image  F  H,  the  same  as  that  formed  by 
the  line  A  C  ;  therefore  a  c  is  the  perspective  of  A  C.  From  this  investiga- 
tion, it  is  evident  that 

If  a  transparent  plane  be  placed  upright  between  the  eye  and  any  given 
object.)  and  a  straight  line  be  drawn  from  any  point  on  the  object  to  the  eye, 
the  point  in  the  plane  pierced  by  the  line  will  be  the  perspective  of  the  point 
from  which  the  line  is  drawn. 


Pig.  7. 


23.    Since  linear  perspective  deals  only  with  the  outlines  of  objects,  it  is 


16 


LINEAR  PERSPECTIVE. 


• 


evidently  unnecessary  to  determine  every  point  in  the  plane  through  which 
the  visual  rays  pass.  If  we  simply  secure  those  points  pierced  by  the  rays 
from  the  prominent  points  of  the  object,  as,  for  example,  those  from  the 
extremities  of  right  lines,  and  those  from  points  in  curved  lines,  which  indi- 
cate their  course,  we  do  all  that  is  needful  to  secure  the  outline  ;  for,  having 
determined  these  points,  we  have  but  to  connect  them  by  lines  corresponding 
to  those  in  the  object,  to  complete  the  drawing.  With  this  understanding, 
let  us  proceed  to  find  the  perspective  of  some  simple  form,  and,  as  the 
square  is  well  suited  to  our  purpose,  we  select  this  for  our  first  trial. 


24.  Let  E  F  D  I,  Fig.  8, 
be  the  square.  Suppose  this 
figure  to  be  traced  on  the 
ground  plane,  which  may  be 
represented  by  the  plane  of 
the  paper  upon  which  the  fig- 
ure is  made. 

On  the  side  F  D  of  the 
square,  suppose  a  transparent 
plane  to  be  erected,  perpen- 
dicular to  the  ground  plane. 
Let  this  plane  be  represented 
by  the  rectangle  M  B  L  N. 
As  we  have  supposed  the 
ground  plane  to  be  repre- 
sented by  the  paper  on  which 
the  figure  is  traced,  the  rec- 
tangle M  B  L  N  must  be  un- 
derstood as  representing  a 
plane  perpendicular  to  the 
plane  of  the  paper.  Suppose 
the  observer  standing  at  A,  a 
point  on  the  same  plane  on 
which  the  square  is  traced, 
and  let  the  eye  be  on  a  level 
with  the  point  C,  a  point  on 
tlie  picture  plane.  This  will 
place  the  eye  vertically  over 
A,  at  a  height  above  the 
ground  plane,  equal  to  the 
height  of  C  above  B  L,  the  base  line. 

By  referring  to  Fig.  9,  the  pupil  will  obtain  a  clear  perception  of  the 


PRINCIPLES  DEVELOPED. 


17 


relation  which  the  planes  sustain  to  each  other,  to  the  square,  and  to  the 
observer.  In  the  two  drawings,  corresponding  points  are  indicated  by  the 
same  letters. 

In  Fig.  8,  the  eye  was  as- 
sumed to  be  above  the  ground 
plane,  at  a  height  equal  to  the 
height  of  the  point  C  above 
the  base  line ;  but  the  pomt  A, 
to  which  the  right  lines  from 
E  and  I  are  drawn,  is  on  the 
same  plane  with  the  square. 
It  may  be  readily  shown,  how- 
ever, that  if  perpendiculars  be 
erected  on  the  points  K  and  J, 
where  these  lines  intersect  the 
base  line  of  the  picture  plane, 
that  lines  drawn  from  the 
points  E  and  I,  to  the  eye,  will 
pass  through  the  picture  plane 
at  points  somewhere  in  these 
perpendiculars.  Take  a  line 
from  E  to  the  eye,  for  example. 
Suppose  a  vertical  plane  erect- 
ed on  the  line  A  E,  equal  in 
height  to  the  assumed  height 
of  the  eye,  a  line  drawn  from 
E  to  the  eye  will  lie  in  this 
plane,  and  the  point  in  the 
picture  plane  through  which  it 
will  pass  will  be  found  some- 
where in  the  line  of  intersec- 
tion of  the  two  planes.  Since 
both  planes  are  vertical,  the 
line  of  their  intersection  will 
be  perpendicular  to  the  base  of 
either  plane.  The  line  K  e  is 
perpendicular  to  B  L,  the  base 

of  the  picture  plane ;  and  therefore  a  line  from  E  to  the  eye,  will  pass 
through  the  plane  at  some  point  in  this  line.  In  the  same  way  it  may  be 
shown  that  a  line  from  I,  to  the  eye,  will  pass  through  the  plane  at  some 
point  in  the  perpendicular  erected  on  the  point  J. 

To  determine  the  points  in  these  perpendiculars  through  which  lines  from 
3 


« 


18 


LINEAR  PERSPECTIVE. 


E  and  I  will  pass,  we  must 
refer  to  Fig,  10.  In  this  M 
drawing,  the  figure  Z  E  A  P 
is  a  representation  of  the  ver- 
tical plane,  supposed  to  be 
erected  on  the  line  A  E. 
The  line  Y  K  is  a  line  formed 
by  the  intersection  of  this 
plane  with  the  picture  plane 
MBLN. 

A  line  drawn  from  E  to  P, 
the  place  of  the  eye,  passes     ^  ^ 
through  Y  K  at  the  point  e. 

Take  the  measure  of  the  line  K  e,  which 
measures  the  height  of  the  point  e  above 
the  base  of  the  plane,  and  lay  it  off  on  the 
perpendiculars  from  the  points  K  and  J, 
Fig.  8,  and  we  determine  e  as  the  point  in 
the  plane  through  which  a  line  from  E,  to 
the  eye,  will  pass,  and  i  as  the  point 
through  which  a  line  from  I  will  pass. 

Since  the  points  F  and  D  are  in  the  base 
line  of  the  picture  plane,  as  well  as  in  the 
front  side  of  the  square,  their  perspectives 
will  appear  in  the  plane  at  the  points  F 
and  D. 


E 

Y 
1 

\                Fig.  8.  / 

\  A 

\g/'' 

k 

PRINCIPLES  DEVELOPED. 


19 


Connect  the  points  now  determined,  and  we  have  the  figure  F  e  i  D,  the 
perspective  of  the  square. 

25.  In  determining  the  perspective  of  the  square,  we  have  developed 
certain  general  truths. 

Examining  the  figure,  we  observe,  first,  that  the  sides  E  F  and  I  D  of  the 
square  are  perpendicular  to  the  picture  plane,  and  that  their  perspectives 
F  e  and  D  on  being  prolonged,  intersect,  or  vanish^  as  it  is  termed,  in  the 
point  C,  which  point  is  the  centre  of  view,  it  being  on  a  level  with,  and 
directly  opposite  the  eye.  It  is  evident  that  what  is  true  of  the  perspectives 
of  E  F  and  I  D  is  also  true  of  the  perspectives  of  all  lines  perpendicular  to 
the  picture  plane ;  and  hence  the  rule :  — 

Rule  1.  The  perspectives  of  all  lines  perpendicular  to  the  picture  plane 
tend  to  the  centre  of  view. 

Upon  a  further  examination  of  the  figure,  we  notice  that  the  sides  F  D 
and  E  I  of  the  square  are  parallel  to  the  picture  plane,  and  that  their  per 
spectives,  F  D  and  e  i,  are  parallel  to  the  lines  they  represent.  It  is  evident 
that  what  is  true  of  the  perspectives  of  F  D  and  E  I  is  also  true  of  the  per- 
spectives of  all  lines  parallel  to  the  picture  plane  ;  and  hence  the  rule :  — 

Rule  2.  The  perspectives  of  all  lines  parallel  to  the  picture  plane  are 
parallel  to  the  lines  they  represent. 


26.  Having  determined  the  perspective  of  tne  square  in  what  is  called 
parallel  perspective,  it  is  now  proposed  to  find  the  perspective  of  the  same 
figure  in  oblique  perspective,  that  is,  with  the  sides  of  the  square  oblique  to 
the  picture  plane. 

Let  the  figure  S  U  0  T,  Fig.  11,  be  the  square,  the  sides  of  which  lie  at 
an  angle  of  forty-five  degrees  with  the  picture  plane.  It  will  be  observed 
that  this  diagram  is  similar  to  Fig-.  8,  with  the  exception  of  the  additional 
lines.  Suppose  this  figure  seen  from  the  same  point  that  the  square  E  F  D  I 
was  seen  from,  that  is,  from  a  point  vertically  over  A,  and  at  a  height  above 
it  equal  to  the  height  of  C  above  B  L,  the  base  line. 

Guided  by  those  laws  to  which  attention  was  called  in  Fig.  8,  we  may 
determine  the  perspective  of  this  figure,  without  resorting  to  the  method 
there  adopted  in  determining  the  perspectives  of  points. 

Since  the  points  S,  TJ,  0,  and  T,  are  in  the  centres  of  the  sides  of  the 
square  E  F  D  I,  it  is  evident  that  their  perspectives  will  be  found  in  the 
perspective  centres  of  the  sides  of  the  figure  e  ij),  the  perspective  of  the 
square  F  E  I  D. 


(20) 


PRINCIPLES  DEVELOPED. 


21 


Let  us  first  determine  the  perspectives  of  the  points  S  and  0.  The  per- 
spectives of  these  points  may  be  determined  by  finding  the  perspective  of  a 
straight  line  connecting  them.  Conceive  a  line  connecting  the  points  S  and 
0,  and  we  have  a  line  passing  through  the  centre  of  the  square  E  F  D  I, 
parallel  to  the  picture  plane.  The  perspective  of  this  line  will  be  parallel 
to  the  line  supposed  to  be  drawn  from  S  to  0,  Rule  2.  To  determine  the 
perspective  centre  of  the  figure  ¥  eiD,  draw  the  diagonals  D  e  and  F  i ;  and 
sr,  the  point  of  their  intersection,  is  the  point  required.  Draw  through  z  a 
line,  as  s  o,  and  we  have  the  perspective  of  the  line  supposed  to  be  drawn 
from  S  to  0,  and  the  point  5,  as  the  perspective  of  S,  and  o,  as  the  perspec- 
tive of  0. 

The  perspective  of  the  point  T  is  already  determined ;  for  since  this  point 
is  in  the  picture  plane,  its  perspective  is  found  where  the  point  is  placed. 
To  complete  the  drawing,  it  only  remains  to  find  the  perspective  of  the 
point  U.  Tliis  point  may  be  determined  by  finding  the  perspective  of  a 
straight  line  connecting  the  points  T  and  U.  Conceive  a  line  connecting 
these  points,  and  we  have  a  line  perpendicular  to  the  picture  plane.  The 
perspective  of  tliis  line  will  tend  to  C,  the  centre  of  view.  Rule  1.  Draw 
a  line  from  T,  the  perspective  of  T,  in  the  direction  of  C,  to  meet  the  line 
e  i,  and  we  have  T  u,  the  perspective  of  the  line,  and  the  point  u,  as  the 
perspective  of  U. 

Connect  the  points  Uy  o,  and  T,  as  shown  in  the  figure,  and  we  com- 
plete the  perspective  of  the  square. 

27.  Before  examining  this  figure,  let  us  draw  the  horizon.  Since  this 
line  always  appears  to  be  on  a  level  with  the  eye,  it  is  evident  that  its  per- 
spective must  pass  through  the  centre  of  view.  Through  the  point  C,  the 
centre  of  view,  draw  a  line,  as  H  R,  to  represent  the  horizon. 

Examining  the  figure,  we  observe  that  the  square  S  U  0  T,  is  bounded 
by  horizontal  lines  lying  at  an  angle  of  forty-five  degrees  with  the  picture 
plane ;  that  the  perspectives  of  the  sides  of  the  square,  on  being  prolonged, 
vanish  in  the  horizon  at  the  points  Y  and  Y' ;  that  these  points  are  equally 
distant  from  C,  the  centre  of  view ;  and  that  their  distance  from  C  equals 
T  A,  a  line  measuring  the  distance  of  the  picture  plane  from  the  observer, 
that  is,  the  distance  of  picture. 

That  the  points  V  and  V  are  at  a  distance  from  C,  equal  to  the  measure  of  the  line  T  A,  may 
be  thus  shown  :  From  the  points  V  and  V,  drop  perpendiculars  to  meet  the  base  line,  as  V  W 
and  Y'  X,  and  from  the  points  W  and  X  draw  lines  parallel  to  S  T  and  0  T,  to  meet  the  perpen- 
dicular T  A,  making  W  A  and  X  A.  The  lines  W  A  and  X  A  form  with  W  X  angles  of  forty- 
five  degrees.  Since  the  angles  T  W  A  and  T  X  A  are  each  forty-five  degrees,  the  distance  from 
T  to  W,  and  from  T  to  X,  is  equal  to  T  A.  But  C  V  and  C  V  are  each  equal  to  T  W  or  T  X» 
and  consequently  to  T  A. 


22 


LINEAR  PERSPECTIVE. 


It  is  evident  that  what  is  true  of  the  perspectives  of  the  sides  of  the 
square  is  also  true  of  the  perspectives  of  all  horizontal  lines  lying  at  an  angle 
of  forty-five  degrees  with  the  picture  plane ;  and  hence  the  rule  :  — ■ 

Rule  3.  The  perspectives  of  all  horizontal  lines  lying  at  an  angle  of 
forty-five  degrees  with  the  picture  plane^  tend  to  a  point  on  the  horizon  at  a 
distance  from  the  centre  of  view  equal  to  the  distance  of  picture. 

Upon  a  further  examination  of  this  figure,  we  notice  that  the  perspectives 
of  those  lines  which  tend  to  the  right,  as  they  recede,  vanish  on  the  right  of 
the  centre  of  view,  and  that  the  perspectives  of  those  which  tend  to  the  left, 
as  they  recede,  vanish  on  the  left  of  this  point.  What  is  true  of  these  lines 
is  evidently  true  of  all  others  similarly  situated ;  and  hence  the  rule :  — - 

Rule  4.  The  perspectives  of  all  horizontal  lines  lying  at  an  angle  of 
forty-five  degrees  with  the  picture  plane,  tend  to  the  right  or  left  of  the  centre 
of  view,  according  as  the  lines  themselves  tend  to  the  right  or  left  of  this 
point. 


28.  From  this  figure,  we  learn  that  the  perspectives  of  all  lines  perpen- 
dicular to(  the  picture  plane,  tend  to  the  same  point ;  that  the  perspectives  of 
all  parallel,  horizontal  lines,  lying  at  an  angle  of  forty-five  degrees  with  the 
picture  plane,  tend  to  the  same  point.  Now,  what  is  true  of  the  perspectives 
of  these  lines  is  evidently  true  of  the  perspectives  of  all  parallel  lines ;  and 
hence  the  rule :  — 

Rule  5.  The  perspectives  of  all  lines  which  o/re  parallel  to  each  other 
tend  to  the  same  point. 


CHAPTER  III. 

PREPARATION  OF   THE   PAPER   FOR   THE  REPRESENTATION 
OF  OBJECTS  BY  THE  USE  OF  VANISHING  POINTS. 

29.  In  Fig.  12,  let  E  J  K  I  represent  a  sheet  of  paper  on  which  it  is  pro- 
posed to  make  a  drawing.  To  prepare  the  paper  for  this  purpose,  draw 
across  it  a  straight,  horizontal  line,  as  B  L,  to  represent  the  base  line  of  the 
picture  plane,  leaving  sufficient  room  below  this  line  to  accommodate  the 
plan  of  the  object,  and  the  elevations,  if  any  are  needed.  The  upper  part 
of  the  paper,  E  B  L  I,  represents  the  picture  plane  ;  the  lower  part  of  the 
paper,  B  J  K  L,  the  ground  plane.  These  planes,  it  will  be  remembered, 
are  supposed  to  be  perpendicular  to  each  other,  the  picture  plane  being 


PREPARATION  OP  PAPER.  23 

vertical,  and  the  ground  plane  horizontal.  The  supposed  relation  of  these 
planes  to  each  other  being  clear  in  the  mind,  it  will  be  understood  that  any 
point  placed  on  the  ground  plane  will  be  near  or  remote  from  the  picture 
plane,  according  to  the  distance  at  which  the  point  is  placed  from  the  line 
B  L.  The  point  A,  for  example,  represents  a  point  on  the  ground  plane, 
and  in  the  picture  plane,  while  the  point  F  represents  a  point  back  of  the 
picture  plane  and  at  a  distance  from  it  equal  to  the  distance  of  F  from  the 
Ime  B  L. 

E,  ,1 


30.  The  place  of  the  eye  determines  the  location  of  the  point  known  as 
the  centre  of  view,  for  this  point  is  always  on  a  level  with,  and  directly  oppo- 
site, the  eye.  It  also  determines  the  place  of  the  horizon,  for  this  line 
always  appears  to  be  on  a  level  with  the  eye.  In  the  drawing  to  be  made,  let 
us  suppose  the  eye  to  be  on  a  level  with,  and  directly  opposite,  the  point  C. 
This  point  will  then  be  the  centre  of  view.  Having  placed  the  point  C, 
draw  through  it  a  very  light,  horizontal  line,  as  H  R,  to  represent  the  hori- 
zon. The  point  C,  it  will  be  observed,  corresponds  to  the  point  C  in  Fi^.  11, 
and  is  the  vanishing  point  of  all  lines  in  the  object  to  be  represented,  situ- 
ated perpendicular  to  the  picture  plane. 

31.  The  plan  of  the  object  may  now  be  drawn  on  the  ground  plane. 
Suppose  the  object  to  be  represented  is  a  building.  The  plan  of  the  building 
should  be  drawn  on  the  right  or  left  of  the  point  C,  at  that  distance  which 
the  building  is  supposed  to  be  on  the  right  or  left  of  the  eye,  and  at  that 
distance  from  the  base  line  which  the  building  is  supposed  to  be  back  of  the 
picture  plane,  and  making  that  angle  with  the  base  line  which  the  building 
is  supposed  to  make  with  this  plane.    The  plan  drawn,  we  may  then  draw 


24  LINEAR  PERSPECTIVE. 

the  elevations  of  the  building  on  the  ground  plane,  where  they  can  be  best 
accommodated,  or  they  may  be  placed  upon  a  separate  sheet  of  paper. 

32.  A  distance  equal  to  the  assumed  distance  of  the  picture  plane  from 
the  eye  may  now  be  laid  off  on  the  horizon  from  the  centre  of  view,  on 
either  side.  Suppose  that  we  assume  the  picture  plane  to  be  at  a  distance 
from  the  eye  equal  to  the  distance  from  C  to  D,  and  place  a  point,  as  D. 
The  point  D,  it  will  be  noticed,  corresponds  to  Y',  in  Fig.  11,  it  being  placed 
on  the  horizon  at  a  distance  from  the  centre  of  view  equal  to  the  distance  of 
picture,  and  is  the  vanishing  point  of  all  horizontal  lines  inclining  to  the 
right  as  they  recede,  which  make  an  angle  of  forty-five  degrees  with  the 
picture  plane. 

33.  The  paper  now  contains  all  the  lines  and  points  required  in  deter- 
mining the  perspective  of  any  object.  They  are  the  base  line,  B  L,  the  hori- 
zon, H  R,  the  centre  of  view,  G,  and  the  point  of  distance,  D. 


CHAPTER  IV. 

THE  PROPER  PLACE  IN  THE  PICTURE  FOR  THE  CENTRE  OF 
VIEW,  AND  THE  PROPER  DISTANCE  TO  ASSUME  AS  THE 
DISTANCE  OF  PICTURE. 

34.  When  the  eye  is  directed  to  any  scene,  the  image  or  picture  formed 
on  the  retina  of  the  eye  is  evidently  circular  in  form ;  and  it  is  also  evident, 
that  the  image  of  the  particular  point  on  which  the  eye  is  fixed,  is  in  the 
centre  of  this  circular  picture.    This  being  the  case,  it  is  obvious  that 

In  assuming  a  point  as  the  centre  of  view,  it  should  be  taken  near  the  centre 
of  the  picture, 

35.  This  rule,  however,  is  intended  to  apply  to  the  fall  and  complete  picture,  rather  than  to 
the  drawing  of  single  objects,  such  as  are  usually  selected  for  first  practice,  the  object  of  which  is 
simply  to  render  the  pupil  familiar  with  the  principles  of  perspective,  and  the  method  of  applying 
them.  These  examples  seldom  contain  more  than  one  object,  and  as  it  is  generally  desirable  that 
this  should  be  represented  as  situated  on  one  side  of  the  observer,  rather  than  directly  before  him, 
it  necessarily  brings  the  centre  of  view  out  of  the  picture. 

36.  A  proper  selection  of  the  distance  of  picture  is  a  matter  of  the 
first  importance,  inasmuch  as  it  is  upon  this  that  the  naturalness  of  the 


CENTRE  OP  VIEW  AND  POINT  OP  DISTANCE. 


25 


drawing  in  a  great  measure  depends.  The  distance  chosen  is  often  so  very 
trifling  that  the  objects  attempted  to  be  represented  couhl  not  be  seen  with- 
out allowing  the  eye  to  wander  from  a  fixed  point,  whicli  is  never  admissil)le. 
No  more  can  be  truthfully  represented  than  can  be  seen  clearly  at  one  view. 
If  we  take  any  object,  as  a  book,  for  example,  and  hold  it  before  the  eye, 
fixing  it  on  its  centre,  it  will  be  found  that  in  order  to  obtain  a  clear  view  of 
all  its  parts,  it  must  be  held  at  a  distance  equal  to  some  four  or  five  times  its 
longest  measurement  taken  from  the  point  on  which  the  eye  is  fixed ;  that 
is,  the  book  must  be  so  held  as  to  be  seen  under  an  angle  of  about  thirty 
degrees. 

Suppose  one  standing  at  B,  Fig.  13,  looking  in  the 
direction  indicated  by  the  line  B  D  ;  the  angle  ABC, 
an  angle  of  thirty  degrees,  may  be  supposed  to  include 
all  that  can  be  clearly  seen.  Take  A  D  as  a  radius, 
and  describe  a  circle,  as  in  Fig.  14,  and  we  have  the 
largest  size  for  a  picture  which  embraces  all  that  is 
seen,  situated  at  the  distance  of  B  D  from  the  observer. 
The  distance  between  the  points  A  and  D  is,  practi- 
cally, one  fourth  the  length  of  the  line  B  D,  the  dis- 
tance of  the  picture  from  the  eye.  From  this  it  is 
evident  that 

The  proper  distance  to  assume  as  the  distance  of 
picture  is  equal  to  at  least  four  times  the  distance  of 
the  most  remote  point  in  the  picture  from  the  centre 
of  view. 


In  our  illustrations,  want  of  space  has  prevented  us  from  assum- 
ing that  distance  of  picture  which  our  rule  demands.  The  pupH  should 
adhere  to  the  rule. 


Pig.  14. 


CHAPTER  V. 

ON  THE  METHOD  OF  DETERMINING  THE  PERSPECTIVES  OF 

POINTS. 

37.  The  principle  on  which  the  perspectives  of  points  are  found,  by  the 
method  we  are  now  explaining,  is  this  :  — 

The  jierspective  of  any  given  point  is  found  somewhere  in  the  indefinite 
perspective  of  a  straight  line  drawn  through  the  point. 
4 


26 


LINEAR  PERSPECTIVE. 


From  this  principle,  it  follows  that 

If  two  lines  be  drawn  through  any  given  pointy  and  their  indefinite  per- 
spectives found,  the  point  of  their  intersection  with  each  other  is  the  perspec- 
tive of  the  point. 

38.  To  avoid  repetition,  we  shall,  in  the  following  problems,  suppose  the  base  line,  the  horizon, 
the  centre  of  view,  and  the  distance  of  picture  given,  as  shown  in  the  accompanying  diagrams. 
These  lines  and  points  will,  in  all  cases,  be  designated  as  follows.  The  base  line  will  be  marked 
B  L,  the  horizon  H  E,,  the  centre  of  view  C,  and  the  point  of  distance  J). 

It  will  be  observed  that,  in  most  cases,  the  centre  of  view  is  taken  near  the  edge  of  the  paper. 
This  is  done  so  as  to  allow  as  great  a  distance  between  the  points  C  and  D  as  the  limited  size  of 
the  page  will  admit. 


PROBLEMS. 
PARALLEL  PERSPECTIVE. 


Prob.  1.  In  Fig.  15.  A  is  a  point  on  the  ground  plane.  It  is  required  to 
find  its  perspective. 


If  R 


39.  Draw  through  the  point  A  a  line,  as  I  A,  perpendicular  to  B  L,  the 
base  line  of  the  picture  plane.  The  point  I  of  this  line  is  in  the  picture 
plane,  and  therefore  its  perspective  is  at  I.  The  perspective  of  the  line 
tends  to  C.  Rule  1.  Connect  the  points  I  and  C,  and  we  have  I  C,  the 
indefinite  perspective  of  I  A. 

Through  the  point  A  draw  a  line,  as  J  A,  making  an  angle  of  forty-five 
degrees  with  B  L,  the  base  line  of  the  picture  plane.  To  determine  the 
inclination  of  this  line,  place  a  point,  as  J,  in  the  base  line,  at  a  distance 
from  I  equal  to  the  distance  of  A  from  I,  and  connect  the  point  with  A. 
The  point  J  of  this  line  is  in  the  picture  plane,  and  therefore  its  perspec- 
tive is  at  J,  The  perspective  of  the  line  tends  to  D.  Rules  3  and  4.  Con- 
nect the  points  J  and  D,  and  we  have  J  D,  the  indefinite  perspective  of  J  A, 
and  the  point  a,  tlie  point  of  its  intersection  with  I  C,  is  the  perspective  of 
A,  the  given  point. 


t 


OBJECTS  IN  PARALLEL  PERSPECTIVE. 


27 


40.  In  this  figure,  more  lines  have  been  used  than  are  absolutely  required.  This  remark 
applies  to  the  line  J  A,  which  is  introduced  simply  for  the  purpose  of  more  clearly  exemplifying 
the  principle  on  which  the  perspectives  of  points  are  found.  The  pupil  will  readily  see  that  we 
have  no  use  for  the  line ;  all  that  is  needed  is  the  point  J,  and  therefore,  in  drawing,  the  line  may 
be  in  all  cases  omitted. 

41.  Pros.  2.  In  Fig.  16,  A  E  F  I  is  a  square  lying'  on  the  ground  plane. 
It  is  required  to  find  its  perspective. 

Find  a  and  /,  the  perspectives  of  the  points  A  and  F,  as  in  Prob.  1. 
The  sides  A  I  and  E  F  of  the  square  are  parallel  to  the  picture  plane ; 


HC 


D  R 


Fig.  16. 


WC/  llO/VC;  U/  C/j  l/JlVy  \j  VJ.  J..       o-j-v^Ajj.  i/iivy  j^jKj^^^^  J  y  ^^^^   -    -  , 

draw  a  line  to  meet  K  C,  parallel  to  E  F,  and  we  have  ef,  the  perspective 
of  E  F,  which  completes  the  drawing  of  the  square. 

42.  Prob.  3.  In  Fig.  17,  A  B  F  I  is  a  rectangle  lying  on  the  ground 
plane.    It  is  required  to  find  its  perspective. 

Determine  the  perspectives  of  the  points  A  and  E,  as  in  Prob.  1. 

The  sides  A  I  and  E  F  of  the  rectangle  are  parallel  to  the  picture  plane ; 
their  perspectives  are  parallel  to  the  lines  themselves.  Rule  2.  From  the 
point  a,  the  perspective  of  A,  draw  a  line  to  meet  J  C,  parallel  to  A  I,  and 
we  have  a  t,  the  perspective  of  A  I.    From  the  point  e,  the  perspective  o? 


28 


LINEAR  PERSPECTIVE. 


E,  draw  a  line  to  meet  J  C,  parallel  to  E  F,  and  we  have  e  /,  the  perspective 
of  E  F,  which  completes  the  drawing  of  the  figure. 

S.    C  P  R 


Pig  17. 


 — —  IE' 

43.  Prob.  4.  In  Fig.  18,  A  is  a  point  on  the  groumd  plane.  It  is 
required  to  find  the  perspective  of  a  point  situated  vertically  over  A,  and  at 
a  height  above  it  equal  to  the  measure  of  the  line  0  P. 


H  C  D  R 


A  Pig.  18. 


■P 

Find  a,  the  perspective  of  A,  as  in  Prob.  1. 

On  the  point  I  erect  a  vertical  line,  as  E  I,  equal  in  height  to  the  measure 
of  0  P.  Conceive  a  line  connecting  the  point  E  with  the  point  to  be  placed 
in  perspective,  and  we  have  a  line  perpendicular  to  the  picture  plane.  The 


OBJECTS  IN  PARALLEL  PERSPECTIVE. 


29 


point  E  of  this  line  is  in  the  picture  plane,  and  therefore  its  perspective  is 
at  E.  The  perspective  of  the  lino  tends  to  C.  Rule  1.  Draw  E  G,  and 
we  have  the  indefinite  perspective  of  the  line. 

Conceive  a  line  connecting  the  point  A  with  the  point  to  be  placed  in  per- 
spective, and  we  have  a  vertical  line,  a  Ihie  parallel  to  the  picture  phine ;  the 
perspective  of  the  line  is  parallel  to  the  line  itself.  Rule  '2.  On  the  jioint  a, 
the  perspective  of  A,  erect  a  vertical  line  to  meet  E  C,  and  o,  tiie  point  of 
intersection,  is  the  perspective  of  the  given  point. 

44.  In  determining  the  perspective  of  the  given  point,  it  will  he  observed 
that  we  have  found  the  perspective  of  a  vertical  line  of  the  measure  of  O  P, 
standing  on  A,  o  a  being  the  perspective  of  the  line.  lu  this  way,  the 
perspective  of  any  vertical  line  may  be  found. 




D  a 

e 

B 

L 

E  K 
A 

E 

Pig.  19. 

AI 
I 

J 

45.  Prob.  5.  In  Fig.  19,  A  E  J  I  is  the  plan  of  a  cube,  situated  on  the 
ground  plane.    It  is  required  to  find  its  perspective. 

In  placing  a  cube  in  perspective,  elevations  are  not  needed.  Since  all  the  faces  of  the  cube  arc 
known  to  be  similar,  the  measurements  of  the  upright  faces  may  be  taken  from  the  plan. 

Find  aej    the  perspective  of  the  plan,  as  in  Prob.  2. 
Find  na,Y  e,o  i,  and  S  j,  the  perspectives  of  the  vertical  edges  of  the 
cube,  as  in  Prob.  4.    See  sections  43  and  44. 


30 


LINEAR  PERSPECTIVE. 


Connect  the  point  n  with  o,  and  V  with  S,  and  the  drawing  of  the  cube  is 
complete.  ^ 

There  are  but  two  faces  of  the  cube  which  are  seen,  viz.,  the  front  face  and  the  left  face.  The 
perspectives  of  the  lines  bounding  these  faces  are  made  full.  The  perspectives  of  the  edges  not 
seen  are  dotted. 


46.  Examining  the  perspective  of  the  cube,  (^Fig.  19,)  we  find  that  the 
front  and  back  face  present  to  the  eye  their  actual  form.  (We  suppose 
the  cube  transparent,  so  that  each  face  may  be  seen.)  These  faces  present 
affront  view,  that  is,  they  face  the  observer.    From  this  it  is  evident,  that 

A  surface  presenting  a  front  view  appears  of  its  actual  form. 

47.  The  perspectives  of  the  edges  presenting  a  front  view  are  in  the  same 
position  as  the  lines  they  represent.  The  perspectives  of  the  vertical  edges 
of  the  cube  are  vertical ;  the  perspectives  of  the  horizontal  edges  which  face 
the  observer  are  horizontal.    From  this  it  is  evident,  that 

Lines  presenting  a  front  view  appear  in  their  actual  position. 

48.  Comparing  the  perspective  of  the  front  face  with  the  perspective  of  the 
back  face,  we  find  that  the  latter  is  every  way  smaller,  the  lines  bounding  it 
being  shorter  than  those  bounding  the  front  face.    From  this  it  is  evident,  that 

Lines  of  equal  length  presenting  a  front  view,  when  seen  at  unequal  dis- 
tances, appear  unequal  in  length,  the  most  distant  appearing  the  shortest  line. 

49.  Again,  we  notice  that  those  faces  seen  obliquely  —  that  is,  those  faces 
which  recede  from  the  eye  —  do  not  appear  of  their  actual  form  ;  they  are 
foreshortened,  as  it  is  termed.  Notice  how  short  the  distance  between  the 
lines  Y  e  and  n  a,  in  the  perspective  of  the  left  face,  and  between  the  lines 
corresponding  to  them  in  the  perspective  of  each  receding  face,  as  compared 
with  the  distance  between  the  lines  n  a  and  o  i,  in  the  perspective  of  the 
front  face.  Comparing  the  perspective  of  the  left  face  with  the  perspective 
of  the  right,  we  find  the  former  foreshortened  more  than  the  latter.  The 
left  face  is  seen  more  obliquely  than  the  right.  From  these  facts  it  is  evi- 
dent, that 

Surfaces  seen  obliquely  are  foreshortened,  and  the  more  obliquely  they  are 
seen  the  more  they  are  foreshortened. 

50.  There  is  another  poinjt  worthy  of  notice  in  the  perspectives  of  the 
receding  faces,  viz.,  the  perspectives  of  the  edges  of  the  cube  situated  above 
the  eye  incline  downward  as  they  recede,  and  the  perspectives  of  the  edges 
below  the  eye  incline  upward  as  they  recede.  The  edges  referred  to  are 
iiorizontal,  and  they  are  seen  obliquely.    From  this  it  is  evident,  that 


OBJECTS  IN  PARALLEL  TERSPECTIVE. 


31 


Horizontal  lines  seen  obliquely,  if  above  the  level  of  the  eye,  appear  to  incline 


downward  as  they  recede,  and  if  below  the  level  of  the  eye,  they  appear  to 
fy^incUne  upward. 

51.  In  our  examination  of  this  figure,  we  have  already  observed  tliat 
horizontal  lines  presenting  a  front  view,  appear  horizontal ;  that  horizontal 
lines  seen  obliquely,  when  situated  above  the  level  of  the  eye,  appear  to 
incline  downward ;  and  when  situated  below  the  level  of  the  eye,  appear  to 
incline  upward.    From  this,  it  is  evident,  that 

All  horizontal  lines  on  a  level  with  the  eye  appear  horizontal. 

52.  We  have  now  reached  a  point  where  it  becomes  desirable  that  the 
pupil  should  commence  drawing  from  the  object  itself.  In  drawing  from 
objects,  the  sketcher  relies  entirely  upon  the  eye  and  the  reason  to  guide  him 
in  determining  their  perspectives.  The  ability  to  recognize  with  readiness 
and  fidelity  the  form*  presented  to  the  eye  by  an  object,  is  the  result  of 
education ;  nothing  but  discipline  can  beget  it ;  and  therefore  the  sooner  the 
practice  of  drawing  from  the  solid  is  commenced,  after''  the  mind  is  made 
acquainted  with  the  principles  of  perspective,  the  better. 

53.  Let  the  pupil  procure  a  cube, 
and  so  place  it  that  it  may  be  seen, 
as  near  as  may  be,  as  represented  in 
Fig.  20  ;  that  is,  let  the  cube  be 
placed  on  the  right,  below  the  level 

•  of  the  eye,  and  let  its  distance  on 
the  right  be  a  little  greater  than  its 
distance  below  the  level  of  the  eye. 
As  a  matter  of  convenience,  we  will 
suppose  the  cube  to  be  lettered  as 
the  drawing  of  the  cube  in  the  figure 
referred  to. 

In  drawing  from  any  object,  the  first  thing  to  be  done  is  to  decide  upon  the 
size  of  the  drawing.  This,  of  course,  may  be  large  or  small,  as  best  suits  the 
wish  of  the  draughtsman.  The  size  of  the  drawing  fixed  upon,  the  question 
arises,  what  part  of  the  object  shall  be  first  represented.  In  drawing  any 
object,  always  begin  with  that  part  which  may,  with  the  greatest  certainty,  be 
described  with  accuracy.  In  this  instance,  the  front  face  of  the  cube  should, 
first  be  drawn.  Tliis  face  presents  a  front  vievj,  and  appears  of  its  actual 
form.  Draw  a  square,  as  A  B  C  D,  and  assume  ^hat  this  figure  represents 
the  front  face  of  the  cube. 

In  drawing  the  outline  of  any  object,  it  should  at  first  be  made  as  light  as  is  consistent  with  dis- 
tmctness,  and'when  the  outline  is  complete,  if  found  to  be  correct,  it  may  be  strengthened. 


Fig.  20. 


32 


LINEAR  PERSPECTIVE. 


The  left  face  of  the  cube  is  seen  obliquely,  and  it  is  foreshortened.  The 
edge  J  H  appears  to  be  on  the  left  of  A  B.  To  make  this  fact  apparent,  lot 
the  pupil  hold  a  rule  between  the  eye  and  the  cube,  so  that  the  edge  of  the 
rule  shall  exactly  coincide  with  the  edge  A  B ;  then  let  him  bring  the  edge 
of  the  rule  to  coincide  with  the  edge  J  H,  and  he  will  find,  in  doing  this, 
that  the  rule  must  be  moved  to  the  left.  Having  recognized  the  fact,  that 
the  edge  J  H  appears  to  be  on  the  left  of  A  B,  the  point  to  be  decided  is, 
what  is  the  apparent  distance  between  these  edges,  compared  with  the  width 
of  the  front  face.  Suppose  it  is  judged  to  be  one  third  the  width  of  the  front 
face ;  then  place  a  point  to  mark  the  apparent  width  of  the  face,  and  draw 
through  it  a  line,  as  J  H,  of  indefinite  length,  parallel  to  A  B. 

The  upper  face  of  the  cube  is  seen  more  obliquely  than  the  left  face,  and 
it  is  foreshortened  more  than  the  latter.  The  edge  J  E  appears  higher  than 
the  edge  A  D,  as  will  be  seen  if  the  rule  is  held  so  as  to  bring  one  of  its 
edges  to  coincide,- first  with  the  edge  A  D,  and  then  with  J  E.  Determine 
the  apparent  distance  between  these  edges,  comparing  it  with  the  height  of 
the  front  face.  Suppose  that  the  edge  J  E  is  found  to  be  above  A  D,  one 
fifth  the  height  of  the  front  face  ;  then  place  a  point  to  mark  its  apparent 
height,  and  through  this  point  draw  a  line,  as  J  E,  to  meet  J  H,  of  indefinite 
length,  parallel  to  A  D. 

Draw  a  line,  as  J  A.  If  the  apparent  width  of  the  left  and  the  upper  face 
of  the  cube  has  been  accurately  determined,  this  line  truly  represents  the 
apparent  length  and  direction  of  the  edge  J  A. 

The  edges  J  H  and  J  E  being  farther  in  the  distance  than  A  B  and  A  D, 
they  appear  shorter  than  A  B  and  A  D.  Make  the  line  corresponding  to 
J  E  a  little  shorter  than  the  perspective  of  the  edge  A  D,  and  draw  a  line, 
as  E  D.  Make  the  line  corresponding  to  J  H  equal  the  perspective  of  the 
edge  J  E,  and  draw  a  line,  as  H  B,  which  completes  the  perspective  of  the 
visible  outline  of  the  cube. 

If  the  cube  were  transparent,  so  that  the  back  face  could  be  seen,  this  face 
would  appear  of  its  actual  form,  for  it  would  present  a  front  view.  To  com- 
plete the  outline  of  the  cube,  place  a  point,  as  F,  on  a  level  witli  the  point 
corresponding  to  H,  and  directly  under  the  point  corresponding  to  E,  and 
draw  lines,  as  H  F,  E  F,  and  F  C. 

The  drawing  should  now  be  carefully  compared  with  the  cube,  and  if  found 
to  be  correct,  its  faint  outline  may  be  made  stronger. 

54.  The  accuracy  of  a  drawing  may  be  determined  by  the  idea  which  it 
conveys  to  the  mind.  As  a  matter  of  course,  no  drawing  can  convey  the 
idea  of  a  cube,  unless  it  truly  represents  its  apparent  form. 

55.  When  the  cube  can  be  readily  and  accurately  represented,  the  draw- 


OBJECTS  IN  PARALLEL  PERSPECTIVE. 


33 


ing  of  simple  rectilinear  forms  should  follow.  Articles  of  lioiisehold  furni- 
ture furnish  excellent  subjects  for  first  ju-actico.  In  drawing  any  object,  first 
make  its  general  outline  ;  tlien  proceed  with  its  details,  taking  them  u})  in  the 
order  of  their  importance.  For  example,  suppose  the  pupil  has  selected  a 
subject,  as  represented  in  Fig-.  21.  Draw,  lirst,  the  main  outline  of  the 
chest,  drawing  the  lines  in  the  same  order  as  they  were  drawn  in  the  cube. 


The  general  outline  of  the  cover  should  next  be  drawn.  The  inclination 
of  any  line  is  most  readily  and  accurately  determined  by  comparing  it  with 
a  horizontal  line  or  with  a  vertical  line.  It  is  often  the  case,  however,  that 
there  are  no  lines  connected  with  the  line  to  be  drawn,  bearing  either  of  these 
relations  to  it ;  when  this  occurs,  a  line  must  be  conceived  to  exist.  In  the 
present  instance,  we  have  the  vertical  line  B  A,  and  the  horizontal  line  D  13, 
with  which  to  compare  the  inclination  of  B  C. 

Next,  draw  the  lines  representing  the  thickness  of  the  boards  of  which  the 
chest  and  cover  are  made.  Then  draw  the  lines  in  the  interior  of  the  box 
and  cover  formed  by  the  meeting  of  the  sides  and  ends ;  then  those  formed 
by  the  meeting  of  the  top  with  the  sides  and  ends  of  the  cover  ;  then  draw 
the  panels.  The  method  adopted  in  drawing  the  chest  will  suggest  the 
proper  course  to  pursue  in  drawing  a 
great  variety  of  objects. 


56.  In  connection  with  drawing  from 
objects,  it  is  a  most  excellent  practice  to 
draw  occasionally  from  copies,  after  the 
following  manner :  Draw  the  object  rep- 
resented in  Fig.  22,  as  it  would  appear 
if  seen  on  the  opposite  side  of  the  ob- 
server, that  is,  on  the  left ;  or  as  it  would 
appear  if  the  eye  were  dropped  to  a  level 
5 


Pig.  22. 


34 


LINEAR  PERSPECTIVE. 


with  the  top ;  or  as  it  would  appear  if  it  were  turned  partly  round,  so  as 
to  have  the  end,  instead  of  the  front,  present  a  front  view.  Practice  of  this 
kind  is  well  adapted  to  test  one's  knowledge  of  principles. 

57.  Prob.  6.  In  Fig.  23,  A  E  J  I  is  the  plan  of  a  cube,  and  also  of  a 
pyramid;  the  base  of  the  pyramid  rests  on  the  upper  face  of  the  cube,  and  the 
height  of  its  vertex  above  its  base  is  equal  to  the  measure  of  one  half  one  side 
its  base.    It  is  required  to  find  the  perspective  of  the  cube  and  pyramid. 

Find  the  perspective  of  the  cube  as  in  Prob.  5.  Find  the  perspective  of 
the  pyramid  as  follows :  Connect  the  point  Z,  the  plan  of  the  vertex,  with 
the  picture  plane  by  a  perpendicular,  as  F  Z.  On  the  point  F  erect  a  vertical 
line,  as  N  F,  equal  in  length  to  the  height  of  the  cube  and  the  height  of  the 
pyramid  taken  together.    The  line  N  P  measures  the  actual  height  of  the 


T 

p 

H  D 

B 

X 

A 

E 

J 

Fig.  23. 

vertex  above  the  ground  plane.  Conceive  a  line  connecting  the  point  N  with 
the  vertex,  and  we  have  a  line  perpendicular  to  the  picture  plane  ;  its  per- 
spective tends  to  C.  Rule  1.  Draw  N  C,  and  we  have  its  indefinite  perspective. 

Conceive  a  line  connecting  the  vertex  of  the  pyramid  with  the  centre  of 
its  base,  and  we  have  a  vertical  line ;  its  perspective  is  vertical.  Rule  2.  The 
perspective  of  the  lower  extremity  of  this  line  is  the  perspective  centre  of  the 


OBJECTS  IN  PARALLFX  PEBSPECTIVE. 


35 


figure  V  o  s  n,  the  perspective  of  the  base  of  the  pyramid,  as  well  as  of  the 
upper  face  of  the  cube.  To  determiue  the  perspective  centre  of  tliis  fijj^ure, 
draw  the  (Jiagoiials  v  s  and  o  n,  and  the  point  the  point  of  their  intorsee- 
tion,  is  the  point  required.  On  the  point  u  erect  a  vertical  line,  as  u  x,  to 
meet  N  C,  and  we  have  the  perspective  of  the  line  conceived  to  connect  the 
vertex  of  the  pyramid  with  the  centre  of  its  base,  and  the  pohit  x,  the  point 
of  its  intersection  with  N  C,  is  the  perspective  of  the  vertex. 

Connect  the  points  v  o  s  and  n  with  x  and  the  drawing  is  complete. 

58.  The  principles  of  perspective  are  more  frequently  violated  in  the 
representation  of  the  pyramid  than  in  the  drawing  of  any  other  form  equally 
simple.  The  error  usually  consists  in  misplacing  the  point  at  the  vertex. 
Suppose  we  are  sketching  a  building  surmounted  by  a  tower,  the  roof  of 
which  is  in  the  form  of  a  pyramid  ;  such,  for  example,  as  we  have  repre- 
sented in  Fig".  24 ;  having  completed  the  outline  of  the  tower,  we  are  about 
to  draw  the  roof.    By  what  method  can  we  determine  its  perspective  ? 

Draw  a  straight  line  from  A  to  B,  and  we  have  a  diagonal  of  the  base,  a 
line  corresponding  to  v  s.  Fig.  23.  The  perspective  of  the  vertex  of  the 
roof  is  directly  over  the  perspective  centre  of  this  line.  The  perspective 
centre  of  this  line  is  a  little  nearer  B  than  A,  (see  Fig.  23,)  owing  to  the 
fact  that  the  extremity  B  is  farther  in  the  distance  than  A.    Determine  the 


c 


Fig.  24. 


36 


LINEAR  PERSPECTIVE. 


perspective  centre  of  the  line,  as  near  as  you  can  judge,  and  place  a  point, 
as  D,  and  on  this  point  erect  a  vertical  line,  as  D  C,  the  apparent  height  of 
the  roof.  Connect  the  point  C  with  the  points  at  the  vertices  of  the  angles 
of  the  base,  as  shown  in  the  drawing,  and  the  outline  of  the  roof  is  complete. 

59.  Prob.  7.  In  Fig.  25,  we  have  the  plan  and  side  elevation  of  a  build- 
ing.   It  is  required  to  find  its  perspective. 

In  the  plan  the  double  lines  represent  the  walls  of  the  building,  and  the  line  S  P  the  ridge  of 
the  roof. 

Find  au  e  i,  the  perspective  of  the  plan,  as  in  Prob.  3. 

Find  the  perspective  of  the  end  wall  on  the  left,  as  follows :  — 

On  the  point  V  erect  a  vertical  line  of  indefinite  length,  and  in  this  line 
place  a  point,  as  W,  at  a  height  above  Y,  equal  to  the  measure  of  W  A',  the 
height  of  the  vertical  sides  of  the  walls. 

Conceive  a  line  drawn  across  the  wall,  as  N'  0',  in  the  elevation,  con- 
necting the  upper  extremities  of  its  vertical  sides,  and  we  have  a  line  perpen- 
dicular to  the  picture  plane ;  its  perspective  tends  to  the  centre  of  view. 
Eule  1.    Draw  W  C,  and  we  have  its  indefinite  perspective. 

The  vertical  sides  of  the  wall  are  parallel  to  the  picture  plane  ;  their  per- 
spectives are  vertical.  Eule  2.  On  the  points  a  and  u  erect  vertical  lines  to 
meet  W  C,  and  we  have  o  a  and  n  u,  the  perspectives  of  the  vertical  sides  of 
the  wall. 

Take  the  measure  of  the  line  K'  T',  the  height  of  the  peak  of  the  gable 
above  the  base  of  the  wall,  and  lay  it  off  on  the  vertical  line  erected  on  V, 
making  V  X  equal  K'  T'.  The  peak  of  the  gable  is  vertically  over  S  in  the 
plan.  Conceive  a  line  connecting  the  point  X  with  the  peak  of  the  gable, 
and  we  have  a  line  perpendicular  to  the  picture  plane  ;  its  perspective  tends 
to  the  centre  of  view.  Rule  1.  Draw  X  C,  and  we  have  its  indefinite  per- 
^  spective. 

Conceive  a  vertical  line  to  be  dropped  from  the  peak  of  the  gable  to  the 
base  of  the  wall,  as  K'  T',  in  the  elevation ;  the  line  passes  through  the 
centre  of  the  wall ;  its  perspective,  therefore,  will  pass  through  the  perspec- 
tive centre  of  the  perspective  of  the  wall ;  the  line  is  parallel  to  the  picture 
plane ;  its  perspective  is  vertical.    Eule  2. 

Draw  the  diagonals  n  a  and  o  u,  and  the  point  of  their  intersection  with 
each  other  is  the  perspective  centre  of  the  wall ;  through  this  point  draw  a 
vertical  line  to  meet  X  C ;  and  ky  the  point  of  their  intersection,  is  the  per- 
spective of  the  peak  of  the  gable. 

The  perspective  of  the  peak  of  the  gable  may  be  found  by  another  method.  The  peak  of  the 
gable  is  vertically  over  the  point  S  ;  its  perspective,  therefore,  will  be  found  in  the  perspective  of 


38 


LINEAR  PERSPECTIVE. 


a  vertical  line  drawn  to  meet  X  C,  erected  on  the  perspective  of  S.  Find  s,  the  perspective  of  S, 
as  in  Prob.  1,  and  on  this  point  erect  a  vertical  line  to  meet  X  C,  and  k,  the  point  of  its  intersec- 
tion with  this  line,  is  the  point  required. 

Connect  the  points  n  and  o  with  k,  and  the  perspective  of  the  wall  is 
complete. 

Find  the  perspective  of  the  end  wall  on  the  right,  bv  the  method  adopted 
in  determining  the  perspective  of  the  end  wall  on  the  left. 

Having  found  the  perspectives  of  the  end  walls,  connect  the  point  o  with 
/,  k  with     and  n  with  c,  and  the  perspective  of  the  building  is  complete. 

60.  Prob.  8.  In  Fig.  26  we  have  the  plan,  the  front  elevation,  and  the 
side  elevation  of  a  building-.    It  is  required  to  find  its  perspective. 

In  the  plan  the  double  lines  represent  the  walls  of  the  building ;  the  open  space  in  the  side  A  I, 
the  door  way  ;  the  single  lines  in  the  sides  A  U  and  A  I,  the  windows ;  the  line  S  T,  the  ridge  of 
the  roof;  and  the  figure  M  the  plan  of  the  chimney  above  the  roof. 

In  this  figure,  for  the  want  of  room,  the  point  of  distance  is  not  represented.  This  point  may 
be  found  by  extending  the  line  of  the  horizon,  and  the  line  Z  i,  the  perspective  of  Z  I,  in  the 
direction  of  their  convergence,  until  they  intersect ;  the  point  of  their  intersection  is  the  point  of 
distance. 

Find  the  perspective  of  the  outline  of  the  building,  as  in  Prob.  7. 

Find  the  perspectives  of  the  windows  in  the  end  wall,  as  follows :  — 

Place  a  point,  as  J,  in  the  vertical  line  erected  on  F,  at  a  height  above  the 
base  line  equal  to  the  height  of  the  lower  line  of  the  windows  above  the 
base  of  the  building ;  also  place  a  point,  as  P,  in  the  same  line,  at  a  height 
above  F  equal  to  the  height  of  the  upper  line  of  the  windows  above  the 
base  of  the  building. 

The  upper  and  lower  lines  of  the  windows  are  perpendicular  to  the  picture 
plane ;  their  perspectives  tend  to  the  centre  of  view.  Rule  1.  From  the 
points  J  and  P,  draw  lines  to  C,  and  we  have  their  indefinite  perspectives. 

Find  the  perspectives  of  the  points  in  the  side  A  U  of  the  plan  which 
mark  the  width  of  the  windows,  as  in  Prob.  1. 

The  vertical  sides  of  the  windows  are  parallel  to  the  picture  plane ;  their 
perspectives  are  vertical.  Rule  2.  On  the  perspectives  of  the  points  which 
mark  the  width  of  the  windows  erect  vertical  lines  to  meet  the  line  P  C, 
and  we  have  the  perspectives  of  the  vertical  sides  of  the  windows,  which 
completes  their  outline. 

In  determining  the  perspectives  of  the  points  which  mark  the  width  of  the  windows,  we  have 
not  drawn  the  perspectives  of  all  the  lines  passing  through  them  to  their  vanishing  points.  By 
the  course  adopted,  a  confusion  of  lines  is  obviated,  and  the  drawing  is  rendered  more  simple. 
It  may  be  well  here  to  say,  that  lines  drawn  simply  for  the  purpose  of  finding  the  perspectives  of 
points,  not  forming  a  part  of  the  perspective  of  the  object,  need  not  be  extended  farther  than 
is  necessary  to  secure  the  points  required. 


40 


LINEAR  PERSPECTIVE. 


Find  the  perspective  of  the  door,  and  the  perspectives  of  the  windows,  in 
the  front  wall,  as  follows :  — 

The  front  windows  being  of  the  same  size,  and  at  the  same  height  above 
the  base  of  the  building,  as  the  end  windows,  the  perspective  height  of  the 
upper  and  lower  lines  of  the  windows  is  determined  by  the  intersection  of 
the  lines  J  C  and  P  C  with  o  a,  the  perspective  of  the  corner  of  the  building. 

The  upper  and  lower  lines  of  the  windows  are  parallel  to  the  picture 
plane ;  their  perspectives  are  parallel  to  the  lines  themselves.  Rule  2.  From 
the  points  v  and  c,  draw  horizontal  lines  across  the  front  wall,  as  c  and  v  6, 
and  we  have  the  indefinite  perspectives  of  the  lines  referred  to,  also  the 
indefinite  perspective  of  the  upper  line  of  the  door,  this  line  being  at  the 
same  height  above  the  base  of  the  building  as  the  upper  line  of  the  windows. 

Find  the  perspectives  of  the  points  in  the  side  A  I  of  the  plan  which  mark 
the  width  of  the  windows,  as  shown  in  the  drawing. 

The  vertical  sides  of  the  windows  and  door  are  parallel  to  the  picture 
plane ;  their  perspectives  are  parallel  to  the  lines  themselves.  Rule  2.  On 
the  perspectives  of  the  points  which  mark  the  width  of  the  windows  and 
door,  erect  vertical  lines  to  meet  c  and  we  have  the  perspectives  of  the 
vertical  sides  of  the  door  and  windows,  which  complete  their  outline. 

* 

In  drawing  the  chimney,  proceed  as  follows :  — 

Find  h  n  and  n  k,  the  perspectives  of  the  lines  h'  n'  and  n'  k\in  the  plan 
of  the  chimney,  as  in  Prob.  3. 

On  the  point  P,  where  the  perpendicular  P  n '  meets  the  base  line,  erect  a 
vertical  line,  as  P  N,  equal  the  height  of  the  chimney. 

The  line  in  the  top  of  the  chimney  over  h'  n'^in  the  plan,  is  perpendicular 
to  the  picture  plane ;  its  perspective  tends  to  the  centre  of  view.  Rule  1. 
Draw  N  C,  and  we  have  its  indefinite  perspective. 

The  vertical  edges  of  the  chimney  are  parallel  to  the  picture  plane ;  their 
perspectives  are  vertical.  Rule  2.  On  the  points  h  and  n  erect  vertical 
lines  to  meet  N  C,  and  the  drawing  of  the  left  face  is  complete. 

The  points  n'  and  k'/m  the  plan,  are  equally  distant  from  the  picture 
plane ;  therefore  the  perspectives  of  the  vertical  edges  of  the  chimney  rest- 
ing on  these  points  are  of  equal  length.  On  the  point  k  erect  a  vertical 
line  equal  to  t  ^,  and  we  have  w  k,  the  perspective  of  the  vertical  edge 
resting  on  A;'. 

Connect  the  points  t  and  and  the  visible  outline  of  the  chimney  is 
complete. 

The  perspective  of  the  outline  formed  by  the  meeting  of  the  roof  and 
chimney  may  be  determined  as  follows  :  — 

The  point  where  the  ridge  of  the  roof  meets  the  left  face  of  the  chimney, 


OBJECTS  IN  PARALLEL  PERSPECTIVE. 


41 


is  vertically  over  r'  in  the  plan ;  the  perspective  of  the  point  will  be  in  tho 
perspective  of  a  vertical  line  erected  on  the  perspective  of  the  point 
Find  r,  the  perspective  of  r\  as  in  Prob.  1. 

The  perspective  of  a  vertical  line  erected  on  r is  parallel  to  the  line  itself, 
for  the  line  is  parallel  to  the  picture  plane.    Rule  2. 

On  the  point  r  erect  a  vertical  hue  to  meet  u  e,  the  perspective  of  the 
ridge,  and  ^,  the  point  of  their  intersection  with  each  other,  is  the  point 
required. 

« 

We  may  find  the  point  y  by  another  method.  Draw  the  diagonals  of  the  figure  xhnt ;  the 
point  formed  by  their  intersection  with  each  other  is  the  perspective  centre  of  the  figure  ;  through 
this  point  draw  a  vertical  line  to  meet  the  line  of  the  ridge,  and  the  point  of  their  intersection  is 
the  point  reqmred. 

Another  method  of  determining  this  point  is  suggested  in  the  drawing. 

The  line  formed  by  the  meeting  of  the  roof  with  the  left  face  of  the  chim- 
ney is  parallel  with  the  receding  sides  of  the  roof,  and  since  the  perspectives 
of  all  lines  parallel  to  each  other  tend  to  the  same  point,  the  perspective 
of  this  line  will  tend  to  the  vanishing  point  of  the  lines  to  which  we  have 
referred.  Rule  5.  Prolong  o  u  and  j  e,  the  perspectives  of  the  receding 
sides  of  the  roof,  as  shown  in  the  drawing,  and  we  obtain  C ',  their  vanishing 
point.  From  C '  draw  a  line  through  y,  to  meet  the  line  t  n,  and  we  have 
y     the  perspective  of  the  line. 

The  line  formed  by  the  meeting  of  the  roof  with  the  front  face  of  the 
chimney  is  horizontal,  and  parallel  to  the  picture  plane ;  its  perspective  is 
horizontal.  Rule  2.  From  the  point  z,  one  extremity  of  its  perspective, 
draw  a  horizontal  line  to  meet  w  k,  and  we  have  z  s,  the  perspective  of  the 
line,  which  completes  the  drawing. 

61.  Prob.  9.  In  Fig.  27  we  have  the  plan,  the  front  elevation,  and  the 
side  elevation  of  the  building  represented  in  Fig.  26,  the  building  being  seen 
from  a  different  point  of  view.  It  is  required  to  find  the  perspective  of  the 
parts  visible. 

In  the  previous  examples,  the  objects  represented  are  situated  some  little  distance  from,  the 
picture  plane ;  in  this  example,  to  simplify  the  drawing,  one  of  the  end  walls  of  the  building  is 
assumed  to  be  in  the  picture  plane. 

The  end  wall  of  the  building  resting  on  the  side  A  K  of  the  plan  being  in 
the  picture  plane,  its  perspective  is  in  all  respects  similar  to  the  outline  of 
the  wall.  Draw  the  perspective  of  the  wall  and  the  perspectives  of  the 
windows,  as  in  the  elevation. 

The  lines  formed  by  the  base  of  the  front  wall,  by  the  eaves,  and  by  the 
ridge  of  the  roof,  are  perpendicular  to  the  picture  plane ;  their  perspectives 
6 


OBJECTS  IN  PARALLEL  PERSPECTIVE. 


43 


tend  to  the  centre  of  view.  Rule  1.  Draw  A  C,  T  C,  and  Z  C,  and  wo 
have  •their  indefinite  perspectives. 

The  vertical  edge  of  the  front  wall  resting  on  E  is  parallel  to  the  picture 
plane ;  its  perspective  is  vertical.  Rule  2.  Find  e,  the  perspective  of  E,  its 
lower  extremity,  as  in  Prob.  1,  and  on  this  point  erect  a  vertical  line  to  meet 
T  C,  and  we  have  e  V,  its  perspective. 

The  side  of  the  roof  parallel  to  that  represented  by  T  Z  is  parallel  to  the 
picture  plane ;  its  perspective  is  parallel  to  tlie  line  it  represents.  Rule  2. 
The  line  to  be  represented  is  parallel  to  T  Z,  we  may,  therefore,  take  this 
line  as  a  guide  in  drawing  its  perspective.  From  the  point  V  draw  a  line  to 
meet  Z  C,  parallel  to  T  Z,  and  we  have  V      the  perspective  of  the  Hne. 

Find  the  perspective  of  the  door,  and  the  perspectives  of  the  windows,  in 
the  front  wall  of  the  building,  as  the  perspectives  of  the  windows  were 
found  in  the  end  wall,  in  Prob.  8. 

Find  the  perspective  of  the  chimney,  as  in  Prob.  8. 

Find  the  perspectives  of  the  lines  formed  by  the  meeting  of  the  roof  and 
chimney,  as  follows :  — 

The  point  where  the  ridge  of  the  roof  meets  the  front  face  of  the  chimney 
is  vertically  over  S ;  the  perspective  of  the  point  will  be  in  the  perspective  of 
a  vertical  line  erected  on  the  perspective  of  S ;  the  perspective  of  a  vertical 
line  is  vertical.  Rule  2.  Find  5,  the  perspective  of  S,  as  in  Prob.  1,  and  on 
this  point  erect  a  vertical  line  to  meet  Z  the  perspective  of  the  ridge ; 
and  6?,  the  point  of  intersection,  is  the  point  required. 

This  point  may  be  found  by  a  different  process.  Draw  the  diagonals  of  the  figure  w '  w  o  o and 
through  the  point  of  their  intersection  with  each  other  draw  a  vertical  line  to  meet  Z  w,  and  we 
obtain     the  point  of  their  intersection. 

We  may  obtain  the  point  by  another  method.  Place  a  point,  as  d,  in  the  line  Z  equally  dis- 
tant from  the  lines  n  ■  n  and  o'  o.   In  sketching,  the  point  is  determined  in  this  way. 

The  line  formed  by  the  meeting  of  the  roof  with  the  front  face  of  the  chim- 
ney, is  parallel  to  the  picture  plane ;  its  perspective  is  parallel  to  the  line 
itself.  Rule  2.  The  line  to  be  represented  is  parallel  to  T  Z,  and  therefore, 
in  drawing  its  perspective,  we  may  be  guided  by  this  Hne,  From  the  point 
d  draw  a  line  to  meet  n'  n  parallel  to  T  Z,  and  we  have  z  the  perspective 
of  the  line. 

The  line  formed  by  the  meeting  of  the  roof  with  the  left  face  of  the  chim- 
ney is  perpendicular  to  the  picture  plane ;  its  perspective  tends  to  the  centre 
of  view.  Rule  1.  From  the  point  z,  one  extremity  of  the  line,  draw  a  line 
to  meet  m '  m,  in  the  direction  of  C,  and  we  have  the  perspective  of  the  line, 
which  completes  the  drawing. 


44 


LINEAR  PERSPECTIVE. 


62.  The  pupil  should  now  attempt  a  sketch  of  some  simple  building, 
seen  in  parallel  perspective.  In  sketching  any  object  as  large  as  a  buiMing, 
we  may  say,  speaking  in  general  terms,  that  the  distance  of  the  draughts- 
man from  the  object  should  be,  at  least,  fifteen  or  twenty  rods,  and  of  course 
as  much  farther  than  this,  as  he  may  desire.   In  delineating  the  general 


outline  of  a  building  of  the  usual  form,  draw  the  lines  in  the  order  indi- 
cated  by  the  numerals  in  Figs.  28  and  29.  The  dotted  lines  in  these 
figures  are  used  as  aids  in  determining  the  perspectives  of  the  peaks  of  the 
gables,  and,  therefore,  in  the  drawing,  they  should  be  made  very  light. 

If  the  pupil  is  not  perfectly  familiar  with  the  truths  to  which  attention 
was  called  in  Sections  46 — 51,  they  should  be  reviewed  before  attempting  a 
sketch. 

OBLIQUE  PERSPECTIVE. 

68.  Prob.  10.  In  Fig.  30,  A  E  I  F  is  the  plan  of  a  cube  situated  on 
the  ground  plane.    It  is  required  to  find  its  perspective. 

The  picture  plane  passes  through  the  vertical  edge  of  the  cube,  resting  on 
A ;  the  perspective  of  this  edge  is  therefore  a  vertical  line  resting  on  A, 
equal  in  length  to  the  measure  of  either  side  of  the  plan.  On  the  point  A 
erect  a  vertical  line  equal  to  A  E,  and  we  have  k  A,  the  perspective  of 
the  edge. 

Find  e,  z,  and /,  the  perspectives  of  the  points  E,  I,  and  F,  as  in  Prob.  1. 

Connect  the  points  A,  e,  z,  and  /,  as  shown  in  the  drawing,  and  we  have 
the  perspective  of  the  lower  face  of  the  cube. 

Prolong  the  perspectives  of  the  sides  of  the  lower  face  of  the  cube  until 
they  intersect  the  horizon,  and  we  obtain  Y  and  D,  their  vanishing  points. 

The  edge  of  the  cube,  situated  over  A  E,  is  parallel  to  A  E ;  its  perspec- 
tive tends  to  Y,  the  vanishing  point  of  A  e,  the  perspective  of  A  E.  Rule  5. 
Draw  k  Y,  and  we  have  its  indefinite  perspective.    The  edge  over  A  F  is 


(45) 


46  LINEAR  PERSPECTIVE. 

parallel  to  A  F ;  its  perspective  tends  to  D,  the  vanishing  point  of  A  /, 
the  perspective  of  A  F.  Rule  5.  Draw  k  D,  and  we  have  its  indefinite 
perspective. 

The  vertical  edges  of  the  cube  are  parallel  to  the  picture  plane ;  their 
perspectives  are  vertical.  Rule  2.  On  the  points  e  and  /,  the  perspectives 
of  the  lower  extremities  of  the  vertical  edges  resting  on  E  and  F,  erect 
vertical  lines  to  meet  the  lines  k  D  and  k  Y,  and  we  have  j  e  and  m  /,  the 
perspectives  of  these  edges. 

The  edge  situated  over  E  I  is  parallel  to  E  I ;  its  perspective  tends  to  D, 
the  vanisliing  point  of  e  i,  the  perspective  of  E  I.  Rule  5.  Draw  a  line 
from 7,  the  perspective  of  one  extremity  of  the  edge,  to  D,  and  we  have  j  D, 
its  indefinite  perspective.  The  edge  over  I  F  is  parallel  to  I  F  ;  its  perspec- 
tive tends  to  Y,  the  vanishing  point  of  i  /,  the  perspective  of  I  F.  Rule  5. 
Draw  a  line  from  the  perspective  of  one  extremity  of  the  edge,  to  Y,  and 
we  have  m  Y,  its  indefinite  perspective. 

Connect  the  point  n,  formed  by  the  intersection  of  the  lines  last  drawn, 
with  i,  and  we  have  n  the  perspective  of  the  vertical  edge  resting  on  In, 
which  completes  the  perspective  of  the  cube. 

64.  The  pupil  should  now  practise  drawing  from  the  cube,  seen  in 
oblique  perspective.  Place  a  cube '  so  as  to  be  seen  as  represented  in 
Fig-.  31 ;  that  is,  place  the  cube  on  a  horizontal  plane,  and  so  arrange  it  that 
the  point  corresponding  to  H,  in  the  drawing,  shall  be  farther  in  the  distance 
than  the  point  corresponding  to  E.  Suppose  the  cube  lettered  as  shown  in 
the  drawing. 


c 

Pig.  31. 


OBJECTS  IN  OBLIQUE  PERSPECTIVE. 


47 


In  drawing  the  cube,  begin  by  making  a  vertical  line,  as  A  C,  of  any 
desired  length,  and  assume  that  this  line  represents  the  edge  A  C. 

The  faces  F  H  C  A,  and  AGED,  are  seen  obliquely,  and  are  therefore 
foreshortened;  the  face  F  H  C  A  is  seen  more  obliquely  than  the  face 
A  C  10  D,  and  is  therefore  foreshortened  more  than  the  latter.  Taking  the 
edge  A  C  as  a  measure,  determine  the  apparent  distance  of  the  edge  F  H 
from  A  C,  and  the  apparent  distance  of  D  E  from  A  C,  measuring  in  the 
direction  of  the  line  R  T,  and  draw  lines  of  hidefniite  length,  as  N  0  and 
P  S,  parallel  to  the  perspective  of  A  C. 

The  edge  C  H,  being  below  the  level  of  the  eye,  appears  to  incline 
upwards.  To  find  the  perspective  of  the  edge  C  H,  conceive  a  horizontal 
line  drawn  from  the  point  H  to  meet  the  edge  A  C,  as  H  B ;  or  hold  a 
string  in  this  position,  so  as  to  cover  the  point  H ;  note  the  size  of  the  angle 
formed  by  the  line  or  string,  with  the  edge  C  H ;  also  note  the  distance 
from  B,  the  point  in  the  edge,  A  C,  where  the  line  or  string  appears  to 
intersect  it,  to  0,  as  compared  with  the  length  of  the  edge  A  C,  and, 
according  to  the  best  of  your  judgment,  place  a  point,  as  H,  in  the  line 
corresponding  to  N  0,  and  draw  a  line,  as  H  C. 

In  like  manner  determine  the  perspective  of  the  point  E,  and  draw  C  E. 
As  the  cube  is  situated,  the  point  E  does  not  appear  to  be  as  high  above  the 
level  of  C  as  the  point  H,  for  the  reason  that  E  is  not  as  far  in  the  dis- 
tance as  H. 

The  edge  F  H  is  farther  in  the  distance  than  A  C,  and  for  this  reason  it 
appears  shorter  than  A  C,  Determine  its  apparent  length,  and  draw  a  line, 
as  FA. 

The  edge  D  E,  being  farther  in  the  distance  than  A  C,  appears  shorter  than 
this  edge,  and  not  being  as  far  in  the  distance  as  F  H,  appears  longer  than 
F  H.    Determine  its  apparent  length,  and  draw  a  line,  as  A  D. 

The  parallel  edges  A  D  and  F  J,  being  seen  obliquely,  appear  to  con- 
verge as  they  recede  from  the  eye.  Bearing  in  mind  that  the  perspec- 
tives of  parallel  lines  tend  to  the  same  point,  draw  a  line,  as  F  J,  of  indefi- 
nite length,  approaching  the  perspective  of  A  D. 

The  parallel  edges  A  F  and  D  J  appear  to  converge  a^  they  recede, 
tending  to  the  same  point  as  A  F  and  C  H.  From  the  point  D  draw  a  line 
to  meet  the  line  last  drawn,  approaching  the  perspective  of  A  F  and  C  H, 
and  we  have  the  perspectives  of  those  faces  which  are  seen. 

The  perspectives  of  the  faces  which  are  not  seen  may  be  determined  as 
follows :  — 

The  edge  J  I  is  farther  in  the  distance  than  F  H,  the  most  distant  of  the 
vertical  edges  which  are  seen  ;  its  perspective,  therefore,  is  shorter  than 
that  of  F  H.  From  the  point  corresponding  to  J  draw  a  vertical  line,  as 
J  I,  shorter  than  the  perspective  of  the  edge  F  H,  and  of  such  a  length  that 


48 


LINEAR  PERSPECTIVE. 


a  line  connecting  the  points  corresponding  to  H  and  I,  will  tend  to  the  same 
point  as  the  perspectives  of  the  edges  F  J,  A  D,  and  C  E. 

Connect  the  points  corresponding  to  H  and  I,  and  the  points  correspond- 
ing to  I  and  E,  and  the  drawing  is  complete. 

65.  When  the  cube  can  be  drawn  with  readiness  and  accuracy,  the  draw- 
ing of  simple  objects  in  oblique  perspective  should  follow.  A  plain,  old- 
fashioned  chair,  such  as  we  have  represented  in  Fig.  32,  will  furnish  tlie 
pupil  with  a  desirable  subject.  In  drawing  a  chair  of  this  description,  first 
draw  an  outline  like  that  in  Fig.  33,  giving  the  general  form  of  the  chair, 


Fig.  32.  Fig.  33. 

as  presented  to  the  eye,  making  the  lines  very  faint ;  guided  by  this  out- 
line, draw  in  the  details. 

The  method  pursued  in  drawing  the  chair  will  suggest  the  proper  course  . 
to  take  in  drawing  a  great  variety  of  objects,  such  as  tables,  stands,  desks, 
and  the  like.  * 

66.  Fig.  32  may  be  taken  as  an  exercise  in  drawing  from  the  copy. 
Draw  the  chair  as  it  would  appear  if  seen  from  a  point  farther  to  the  right, 
or  left,  than  that  here  taken  ;  or  as  it  would  appear  if  the  back  legs  rested 
on  the  points  E  and  F ;  or  on  the  points  E  and  I ;  or  on  the  points  I  and  J. 

6T.  Prob.  11.  In  Fig.  31  ive  have  the  plan  and  elevations  of  the  build- 
ing represented  in  Figs.  26  and  27,  the  building  being  seen  from  a  different 
point  of  view.    It  is  required  to  find  its  perspective. 


50  LINEAR  PERSPECTIVE.  (J 

The  perspective  of  the  end  wall,  resting  on  the  side  A  U  of  the  plan,  may 
be  found  as  follows. 

The  vertical  corner  of  the  house,  resting  on  the  point  A,  is  in  the  picture 
plane ;  its  perspective,  therefore,  is  a  vertical  line  resting  on  A,  equal  in 
length  to  the  line  itself.  On  the  point  A  erect  a  vertical  line  equal  to  0'  U', 
and  we  have  F  A,  the  perspective  of  the  vertical  side  of  the  wall,  rest- 
ing on  A. 

Find  the  perspective  of  U,  as  in  Prob.  1,  and  connect  ^this  point  with 
A,  and  we  have  A     the  perspective  of  A  U,  the  base  of  the  wall. 

Prolong  the  line  A  to  meet  the  horizon,  and  we  obtain  V,  its  vanishing 
point. 

Conceive  a  line  drawn  across  the  wall,  as  0'  F',  in  the  elevation,  connect- 
ing  the  upper  extremities  of  its  vertical  *sides,  and  we  have  a  line  parallel  to 
the  base  of  the  wall ;  its  perspective  tends  to  V.  Rule  5.  From  the  point 
F  draw  a  line  to  V,  and  we  have  F  Y,  its  indefinite  perspective. 

The  vertical  side  of  the  wall  resting  on  U  is  parallel  to  the  picture  plane  ; 
its  perspective  is  vertical.  Pule  2.  On  the  point  u  erect  a  vertical  line  to 
meet  F  Y,  and  we  have    o,  its  perspective. 

Prolong  the  vertical  line  F  A,  as  shown  in  the  drawing,  making  M  A  equal 
the  height  of  the  peak  of  the  gable  above  the  base  of  the  wall. 

Conceive  a  line  drawn  through  K',  the  peak  bf  the  gable,  parallel  to  U'  A', 
the  base  of  the  wall ;  its  perspective  tends  to  Y,  the  vanishing  point  of  A 
the  perspective  of  the  base  of  the  wall.    Rule  5.    Draw  M  Y,  and  we  have 
its  indefinite  perspective. 

The  peak  of  the  gable  is  vertically  over  the  point  S,  in  the  plan ;  its  per- 
spective is  a  point  in  the  line  M  Y,  where  the  perspective  of  a  vertical  line, 
which  we  may  conceive  to  be  erected  on  S,  will  intersect  this  line.  The 
perspective  of  a  vertical  line  is  vertical.  Rule  2.  Find  5,  the  perspective  of 
S,  as  in  Prob.  1,  with  this  exception ;  omit  drawing  the  indefinite  perspec- 
tive of  c  S.  This  line  may  be  dispensed  with ;  for  the  point  S,  being  in  the 
line  A  U,  its  perspective  must  be  in  A  the  perspective  of  this  line  ;  and 
since  it  is  in  A  it  must  be  at  5,  where  the  indefinite  perspective  of  E'  S 
intersects  this  line.  On  the  point  s  erect  a  vertical  line  to  meet  M  Y,  and  A:, 
the  point  of  their  intersection,  is  the  perspective  of  the  peak  of  the  gable. 

Connect  the  point  o  with  A;,  and  k  with  F,  and  the  perspective  of  the  out- 
line of  the  wall  is  complete. 

To  find  the  perspectives  of  the  windows  in  this  wall,  proceed  as  follows  :  — 

In  the  vertical  line  erected  on  A,  place  a  point,  as  Q,  at  a  height  above  A 
equal  to  the  height  of  the  lower  line  in  the  windows,  above  the  base  of  the 
building ;  also  place  a  point,  as  G,  in  the  same  line,  at  a  height  above  A 
equal  to  the  height  of  the  upper  line  in  the  windows  above  the  base  of  the 
building. 


OBJECTS  IN  OBLIQUE  PERSPECTIVE. 


51 


The  upper  and  lower  lines  in  the  windows  are  parallel  to  the  base  line  of 
the  wall ;  their  perspectives  vanish  at  V,  the  vanishing  point  of  A  u.  Rule  5. 
From  the  points  Q  and  G  draw  lines  to  meet  o  u,  in  the  direction  of  V,  and 
we  have  the  indefinite  perspectives  of  the  upper  and  lower  lines  of  the 
windows. 

The  perspectives  of  the  vertical  sides  of  the  windows  arc  vertical.  Rule  2. 
Find  the  perspectives  of  the  points  in  the  plan  which  mark  the  width  of  the 
windows,  as  the  perspective  of  the  point  S  was  found,  and  on  these  points 
erect  vertical  lines  to  meet  G  F,  and  we  complete  the  perspectives  of  the 
windows. 

The  perspective  of  the  front  wall  may  be  determined  by  a  process  entirely 
similar  to  that  by  which  the  perspective  of  the  end  wall  was  found ;  but 
since  the  vanishing  point  of  the  line  A  I  is  without  the  limits  of  our  page, 
we  must  find  its  perspective  by  another  method. 

Find  ^,  the  perspective  of  I,  as  in  Prob.  1. 

Prolong  the  side  I  E  of  the  plan  to  meet  the  picture  plane,  as  shown  in 
the  drawing,  and  on  Y,  the  point  of  its  intersection  with  the  base  line,  erect 
a  vertical  line  of  indefinite  length,  and  in  this  line  place  a  point,  as  W,  at  a 
height  above  Y  equal  to  the  height  of  the  wall. 

Conceive  a  line  connecting  the  upper  extremity  of  the  vertical  side  of  the 
wall  resting  on  I,  with  W,  and  we  have  a  line  parallel  to  Y  I ;  its  perspec- 
tive tends  to  the  vanishing  point  of  this  line.  Rule  5.  The  line  Y  I  being 
parallel  to  the  line  A  U,  its  vanishing  point  is  V.  Draw  a  line  connecting 
the  points  W  and  V,  and  we  have  W  V,  the  indefinite  perspective  of  the  line 
connecting  the  points  referred  to. 

The  perspective  of  the  vertical  side  of  the  wall  resting  on  I,  is  vertical. 
Rule  2.  On  the  point  i,  the  perspective  of  I,  erect  a  vertical  line  to  meet 
W  Y,  and  we  have  i  b,  the  perspective  of  this  side  of  the  wall. 

Connect  the  point  F  with  b,  and  A  with  i,  and  the  outline  of  the  wall  is 
complete. 

To  find  the  perspective  of  the  door,  and  the  perspectives  of  the  windows, 
proceed  as  follows :  — 

In  the  vertical  line  erected  on  Y  place  a  point,  as  H',  at  a  height  above  Y 
equal  to  the  height  of  the  lower  line  in  the  windows  above  the  base  line  of 
the  wall ;  also  place  a  point,  as  L',  in  this  line,  at  a  height  above  Y  equal  to 
the  height  of  the  upper  line  in  the  windows  and  door  above  the  base  of  the 
building. 

Conceive  the  lower  line  in  the  windows,  and  the  upper  Hue  in  the  door 
and  windows,  prolonged  to  meet  the  vertical  sides  of  the  wall,  as  shown  in 
the  elevation.    Conceive  a  line  connecting  the  point  L'  with  a  point  in  the 


52 


LINEAR  PERSPECTIVE. 


vertical  side  of  the  wall  resting  on  I,  corresponding  to  D',  in  the  elevation  ; 
also  conceive  a  line  connecting  the  point  H'  with  a  point  in  this  side  of  the 
wall  corresponding  to  B'  in  the  elevation.  These  lines  are  parallel  to  that 
represented  hjWb;  their  perspectives  vanish  at  V.  Rule  5.  From  the 
points  L'  and  draw  lines  to  meet  b  i,  in  the  direction  of  V,  and  we  obtain 
/  and  /,  the  perspectives  of  points  in  the  vertical  side  of  the  wall  resting  on 
I,  corresponding  to  B'  and  B',  in  the  elevation. 

The  points  G  and  Q,  in  the  perspective  of  the  vertical  side  of  the  wall 
resting  on  A,  correspond  to  G'  and  QS  in  the  elevation.  Connect  the  point 
G  with  /,  and  Q  with  /,  and  we  have  G  /  and  Q  the  indefinite  perspectives 
of  the  upper  and  lower  hues  of  the  windows,  and  the  upper  line  of  the 
door. 

From  the  points  in  the  side  A  I  of  the  plan,  which  mark  the  width  of  the 
door  and  windows,  draw  lines  to  meet  the  base  line  of  the  picture  plane, 
parallel  to  the  side  A  U  of  the  plan.  The  perspectives  of  these  lines  tend 
to  Y.  Rule  5.  From  the  points  where  these  lines  intersect  the  base  line, 
draw  lines  to  meet  A  i,  in  the  direction  of  V,  and  the  points  of  their  inter- 
section with  A  i  are  the  perspectives  of  the  points  in  the  side  A  I  of  the 
plan  which  mark  the  width  of  the  door  and  windows.  On  the  points  thus 
determined,  erect  vertical  lines  to  meet  G  /,  and  we  complete  the  drawing 
of  the  door  and  windows. 

The  perspective  of  the  roof  may  be  found  as  follows :  — 

We  have  already  found  k  F  and  F  b,  the  perspectives  of  two  sides  of  the  roof,  and  to  complete 
its  outline,  we  have  only  to  findj,  the  perspective  of  the  end  of  the  ridge  over  T,  in  the  plan,  and 
to  connect  this  point  with  7i  and  b. 

In  the  vertical  line  erected  on  Y  place  a  point,  as  X,  at  a  height  above  Y 
equal  to  the  height  of  the  ridge  of  the  roof  above  the  base  of  the  building. 

Conceive  a  line  connecting  the  end  of  the  ridge  over  T  with  the  point  X, 
and  we  have  a  line  parallel  to  Y  E  ;  its  perspective  tends  to  Y,  the  vanishing 
point  of  this  line.  Rule  5.  Draw  a  line  from  X  to  Y,  and  we  have  the 
indefinite  perspective  of  the  line  conceived  to  connect  the  end  of  the  ridge 
with  X. 

The  end  of  the  ridge  is  vertically  over  T  ;  its  perspective  is  a  point  in  the 
line  drawn  from  X  to  Y  where  the  perspective  of  a  vertical  line,  which  he 
may  conceive  to  be  erected  on  T,  will  intersect  this  line.  The  perspective 
of  a  vertical  line  is  vertical.  Rule  2.  Find  t,  the  perspective  of  T,  as  in 
Frob.  1,  and  on  this  point  erect  a  vertical  line  to  meet  X  Y ;  and/,  the  point 
of  their  intersection,  is  the  perspective  of  the  end  of  the  ridge  over  T. 

Connect  the  point  k  with  j,  and  /  with  and  the  outline  of  the  roof  is 
complete. 


O^ECTS  IN  OBLIQUE  PERSPECTIVE. 


63 


The  perspective  of  the  chimney  may  be  found  as  follows  :  

Prolong  the  lines  n'  h'  and  d'  x' ,  in  the  plan  of  the  chimney,  to  meet  the 
base  line  of  the  picture  plane.  The  lines  N  h'  and  J  x'  are  parallel  to  A  U ; 
their  perspectives  tend  to  V.  Rule  5.  Draw  N  V  and  J  V,  and  wc  have 
their  indefinite  perspectives. 

On  the  points  N  and  J  erect  vertical  lines,  as  N  N'  and  J  J',  equal  to  the 
height  of  the  chimney. 

Conceive  the  upper  lines  of  the  chimney  parallel  to  n'  h'  and  d'  x'  pro- 
longed to  meet  the  picture  plane ;  they  intersect  the  plane  at  N'  and  J' ; 
their  perspectives  tend  to  V,  the  vanishing  pohit  of  the  perspectives  of  N  A' 
and  J  x'o  Rule  5.  Draw  hues  from  N'  and  J'  to  V,  and  we  have  their 
indefinite  perspectives. 

Find  h,  n,  and  d,  the  perspectives  of  h'  n'  and  d',  in  the  plan,  as  the  per- 
spective of  the  point  S  was  found. 

The  perspectives  of  the  vertical  edges  of  the  chimney  are  vertical. 
Rule  2.  On  the  points  A,  n,  and  d,  erect  vertical  hues  to  meet  N'  V  and 
J'  Y,  and  we  have  the  perspectives  of  the  vertical  edges  which  are  seen. 

Connect  the  points  a  and  iv,  and  the  drawing  of  the  chimney  is  complete. 

The  perspective  of  the  outline  formed  by  the  meeting  of  the  roof  with 
the  chimney  may  be  found  as  follows :  — 

The  point  where  the  ridge  of  the  roof  meets  the  left  ^ce  of  the  chimney 
is  vertically  over  r ' ;  the  point  where  the  ridge  of  the  roof  meets  the  right 
face  of  the  chimney  is  vertically  over  v ' ;  the  perspectives  of  these  points 
will  be  in  the  line  kj^  the  perspective  of  the  ridge,  where  the  perspectives  of 
vertical  lines,  which  we  may  conceive  to  bo  erected  on  r '  and  v will  inter- 
sect this  line.  The  perspectives  of  vertical  lines  are  vertical.  Rule  2.  To 
find  the  perspectives  of  r'  and  v\  draw  a  line  from  5,  the  perspective  of  S, 
to  the  perspective  of  T  ;  the  point  r,  where  s  t  intersects  N  C,  is  the  per- 
spective of  r and  the  point  where  it  intersects  J  C,  is  the  perspective  of  v 
On  the  points  r  and  v  erect  vertical  lines  to  meet  k  y,  and  we  obtain  e  and 
m,  the  perspectives  of  the  points  referred  to. 

The  lines  formed  by  the  meeting  of  the  roof  with  the  left  and  right  faces 
of  the  chimney  are  parallel  to  the  sides  of  the  roof,  represented  by  F  k  and 
b  j  ;  their  perspectives  tend  to  the  vanishing  point  of  these  lines.  Rule  5. 
Prolong  the  lines  F  k  and  b  j  in  the  direction  of  their  convergence,  until 
they  intersect  each  other,  and  from  the  point  of  their  intersection  draw  a 
line  through  e  to  meet  the  line  a  and  a  line  through  m  to  meet  the  line 
w  d,  and  we  have  e  z  and  m  the  perspectives  of  the  lines  formed  by  the 
meeting  of  the  roof  with  the  left  and  right  faces  of  the  chimney. 

If  the  pupil  has  been  accurate  in  his  drawing,  the  lines  F  k  and  bj  intersect  at  a  poiiit 
vertically  over  V. 


64 


LINEAR  PERSPECTIVE.  .« 


Connect  the  points  z  and  and  we  have  the  perspective  of  the  line  formed 
by  the  meeting  of  the  roof  with  the  front  face  of  the  chimney,  which  com- 
pletes the  drawing. 

68.  In  sketching  a  building  seen  in  oblique  perspective,  of  the  form  rep- 
resented in  Figs.  35  and  36,  draw  the  lines  in  the  order  indicated  by  the 


Fig.  35.  Fig.  36. 


numerals.  The  dotted  lines  in  these  figures,  which  are  not  used  in  deter- 
mining the  perspectives  of  the  peaks  of  the  gables,  represent  imaginary 
lines.  The  remarks  made  on  the  line  H  B,  in  Fig.  31,  will  suggest  the 
use  of  these  lines.  • 

OBJECTS  CONTAINING  CURVED  LINES. 

69.  Prob.  12.  In  Fig.  37,  NIMK  is  a  circle  lying  on  the  ground 
plane.    It  is  required  to  find  its  perspective. 

The  perspective  of  a  circle  is  determined  by  finding  the  perspectives 
of  a  number  of  points  in  its  circumference,  and  drawing  a  curved  line 
through  them.  The  number  of  points  usually  taken  is  eight,  and  they 
are  determined  as  follows  :  — 

Dl-aw  a  square  enclosing  the  circle,  as  A  F  E  J,  making  the  side  A  J  par- 
allel with  the  base  line  of  the  picture  plane. 

Draw  the  diagonals  A  E  and  F  J,  and  through  the  point  of  their  intersec- 
tion with  each  other  draw  lines,  as  I  K  and  N  M,  parallel  with  the  sides  of 
the  square. 

Through  the  points  where  the  diagonals  intersect  the  circumference  of  the 
circle,  draw  lines,  as  shown  in  the  figure. 

We  now  have  eight  points  in  the  circumference  of  the  circle,  equally  dis- 
tant from  each  other,  viz.,  N,  0, 1,  Y,  M,  U,  K,  and  S.  The  perspectives  of 
these  points  may  be  found  as  follows :  — 


V 


OBJECTS  CONTAINING  CURVED  LINES.  65 

Find  afej,  the  perspective  of  the  square  A  F  E  J,  as  in  Prob.  2. 
Draw  a  e  and  /  y,  and  we  have  the  perspectives  of  the  diagonals  A  E 
and  F  J. 

Tlic  line  N  M  is  parallel  to  the  picture  plane ;  its  perspective  is  parallel  to 
N  M.  Rule  2.  Through  c,  the  perspective  of  C,  draw  a  line  to  meet  W  C 
and  P  C,  parallel  to  N  M,  and  we  have  w,  the  perspective  of  N,  and  w,  the 
perspective  of  M. 

Prolong  the  lines  0  S,  I  K,  and  V  U,  to  meet  the  base  of  the  picture 
plane ;  these  lines  are  perpendicular  to  the  picture  plane  ;  their  perspectives 
tend  to  C.    Rule  1.    Draw  X  C,  Y  C,  and  Z  C,  and  we  have  their  indefinite 


perspectives.  The  points  o,  s,  and  v,  formed  by  the  intersection  of  the 
lines  X  C  and  Z  C  with  a  e  and  /  j,  the  perspective?  of  the  diagonals,  are 
the  perspectives  of  the  points  0,  S,  U,  and  Y.  The  points  i  and  k,  formed 
by  the  intersection  of  the  line  Y  C  with  a  j  and  f  e,  the  perspectives  of  the 
sides  A  J  and  F  E  of  the  square,  are  the  perspectives  of  the  points  I  and  K. 

Through  the  points  n,  o,  t,  v,  m,  w,  and  s,  draw  a  curved  line,  as  shown 
in  the  figure,  and  we  have  the  perspective  of  the  circle. 


70.  In  Fig.  37,  the  eight  points,  N,  0, 1,  &c.,  in  the  circumference  of  the 
circle,  are  equidistant  from  each  other ;  if  these  points  be  properly  connected 


56  LINEAR  PERSPECTIVE. 

by  straight  lines,  the  figure  formed  is  an  octagon,  and  if  the  perspectives  of 
these  points  be  properly  connected  by  straight  lines,  the  figure  formed  is  the 
perspective  of  the  octagon. 

71.  Examining  the  perspective  of  the  circle,  we  observe  that  the  per- 
spective of  the  half  nearest  the  eye  is  larger  than  that  of  the  half  most  dis- 
tant, the  distance  from  i  to  c  being  greater  than  from  c  to  k.  Examining 
the  perspective  of  the  circumference  of  the  circle,  we  notice  that  no  part  of 
the  line  is  the  arc  of  a  circle,  the  figure  described  by  it  being  that  of  an 
ellipse. 

To  represent  a  circle  seen  obliquely  is  more  difficult  than  the  drawing  of  any  other  simple 
form,  and  it  will  be  well  for  the  pupil  to  make  a  careful  study  of  the  perspective  of  this  figure 
before  he  attempts  to  represent  an  f>bject  containing  curved  lines. 

72.  In  sketching,  the  perspective  of  the  circle  may  be  determined  as 
follows  :  Suppose,  for  example,  that  you  are  about  to  draw  the  outline  of  a 
wheel,  seen  as  the  circle  N  I  M  K,  in  Fig.  37,  when  the  page  lies  horizon- 
tally before  you,  and  you  are  looking  in  the  direction  of  the  line  K  I. 

Draw,  first,  a  light,  horizontal  line,  as 
a  6,  in  Fig.  38,  and  assume  that  this  line 
represents  the  distance  between  the  points 
in  the  rim  of  the  wheel  corresponding  to 
N  and  M  in  the  circle. 

Determine  the  apparent  distance  be- 
tween the  points  in  the  rim  of  the  wheel, 
corresponding  to  K  and  I  in  the  circle,  as  compared  with  the  distance 
between  the  points  corresponding  to  N  and  M,  and  place  points,  as  i  and  e, 
Fig.  38,  opposite  the  centre  of  the  line  a  b,  to  represent  the  apparent  dis- 
tance between  the  points  referred  to,  making  the  distance  from  ^  to  c  a  little 
less  than  from  e  to  c. 

Guided  by  the  line  a  b,  draw  through  the  points  a,  i,  b,  and  e,  the  per- 
spective of  the  circle,  making  the  line,  at  first,  very  light.  In  drawing  this 
line,  begin  with  that  part  which,  when  drawn^will  not  be  covered  by  the 
pencil  or  hand  in  drawing  the  remainder  of  the  line.  The  perspective  of  a 
circle,  seen  obliquely,  in  any  position,  may  be  drawn  in  a  similar  manner. 

73.  Prob.  13.  In  Fig.  39,  the  circle  N  I  M  K  i5  the  plan  of  a  cylinder; 
the  lower  base  rests  on  the  ground  plane ;  the  height  of  the  cylinder  is  equal 
to  C  M,  one  half  the  diameter  of  the  circle.  It  is  required  to  find  its  per- 
spective. 

Find  the  perspective  of  the  lower  base  of  the  cylinder,  as  in  Prob.  12. 


OBJECTS  CONTAINING  CURVED  LINES. 


57 


To  find  the  perspective  of  the  upper  base,  proceed  as  follows :  — 
On  the  points  Q  and  P  erect  vertical  lines  equal  to  C  M,  the  height  of 
the  cylinder,  and  connect  the  points  T  and  G. 

In  the  line  T  G  place  a  point,  as  ^V,  vertically  over  W ;  a  point,  as  X', 
vertically  over  X ;  a  point,  as  Y',  vertically  over  Y ;  and  a  point,  as  Z',  ver- 
tically over  Z. 

Conceive  the  upper  base  of  the  cylinder  circumscribed  by  a  square,  and 
lines  drawn  across  it,  as  shown  in  the  plan,  and  these  lines  extended  to  meet 
the  picture  plane.  The  line  corresponding  to  W  P  meets  the  plane  at  W', 
the  line  corresponding  to  X  S  meets  the  plane  at  X',  the  line  corresponding 
to  Y  K  meets  the  plane  at  Y',  the  line  corresponding  to  Z  U  meets  the  plane 
at  Z',  the  line  corresponding  to  P  E  meets  the  plane  at  G,  and  the  line  cor- 
responding to  Q  E  meets  the  plane  at  T.  The  lines  meeting  the  plane  at 
W,  X',  Y',  Z',  and  G,  are  perpendicular  to  the  picture  plane  ;  their  perspec- 
tives tend  to  C.  Rule  1.  Draw  W  C,  X'  C,  Y'  C,  Z'  C,  and  G  C,  and  we 
have  their  indefinite  perspectives.  The  line  meeting  the  plane  at  T  is  par- 
allel to  Q  E ;  its  perspective  tends  to  D.   Rule  3, 


S  C  D  R 


To  complete  the  drawing  of  the  upper  base,  proceed  as  in  drawing  the 
lower  base. 

Connect  the  point  r  with  w,  and  d  with  w,  and  the  drawing  is  complete. 

8 


58 


LINEAR  PERSPECTIVE. 


74.  In  sketching  any  cylindrical  object,  as  a  cask,  for  example,  to  de- 
termine its  general  outline,  first  draw  a  rectangle,  as  A  B  C  D,  in  Fig.  40, 
giving  the  apparent  height  of  the  cask,  and  its  width  through  the  chimes. 
Guided  by  the  line  A  D,  draw  the  ellipse  as  in  Fig.  38,  forming  the  perspec- 
tive outline  of  the  upper  head ;  guided  by  the  lines  A  B  and  D  C,  draw  the 
outlines  of  the  upright  sides ;  and  guided  by  the  line  B  C,  draw  what  is 
seen  of  the  outline  of  the  lower  head. 


1 


Fig.  40. 


QUESTIONS. 


1.  What  instruction  is  given  on  the  selection  of  paper?  Why  should  the  paper 
be  placed  upon  a  hard  surface  in  drawing?    What  is  the  objection  to  rolling  paper? 

2.  What  are  the  desirable  qualities  of  a  lead  pencil  ?  Whose  pencils  are  recom- 
mended ?  How  are  Faber's  pencils  marked  ?  What  pencils  are  suitable  for  draw- 
ing diagrams  ?  What  pencils  are  suitable  for  sketching  ?  What  instruction  is  given 
on  sharpening  the  pencil? 

3.  What  are  the  desirable  qualities  of  rubber  ?  Whose  rubber  is  recommended  ? 
What  is  said  on  the  use  of  rubber  ?  How  are  you  to  avoid  the  necessity  of  too  fre- 
quently resorting  to  its  use  ? 

4.  What  is  said  of  the  rule  ? 

5.  For  what  is  the  right-angled  triangle  used  ?  What  size  is  most  convenient  ? 
Upon  what  does  the  value  of  the  instrument  depend  ?  How  may  the  accuracy  of 
the  right  angle  be  tested  ?  How  is  this  instrument  used  in  drawing  perpendiculars  ? 
How  is  it  used  in  drawing  parallels  ? 

6.  What  is  said  of  the  compasses?  To  what  use  is  this  instrument  applied? 
When  working  with  the  compasses,  how  should  they  be  held  ? 

CHAPTER  I. 
Of  what  does  this  chapter  treat  ? 

7.  What  is  linear  perspective  ?  What  is  the  perspective  of  an  object  ?  What  is 
the  perspective  of  a  line  or  point  ?  When  the  perspective  of  a  line  is  indefinitely 
prolonged,  what  is  the  line  called  ? 

8.  What  is  a  plane  ?  May  we  assume  any  plane  to  be  extended  indefinitely  in 
»  any  required  direction  ? 

9.  What  is  the  ground  plane  ?  What  is  the  position  of  the  ground  plane  ?  What 
part  of  Fig  1  represents  this  plane  ? 

10.  What  is  the  picture  plane  ?  What  is  the  position  of  the  picture  plane?  How 
is  it  situated  with  regard  to  the  observer  ?    What  part  of  Fig.  1  represents  this  plane  ? 

11.  What  is  the  point  of  view  ?    What  represents  this  point  in  Fig.  1  ? 

12.  What  is  the  centre  of  view  ?    What  represents  this  point  in  Fig.  1  ? 

13.  What  is  said  of  the  horizon?    Does  the  horizon  under  all  circumstances 

(59) 


60 


LINEAR  PERSPECTIVE. 


appear  to  be  on  a  level  with  the  eye  ?  What  is  the  perspective  of  the  horizon  called  ? 
What  represents  the  horizon  in  Fig.  1  ? 

14.  What  is  the  distance  of  picture  ?  What  line  in  Fig.- 1  measures  the  distance 
of  picture? 

15.  What  is  the  point  of  distance  ? 

16.  What  are  visual  rays  ?    How  are  visual  rays  represented  ? 

17.  What  is  a  vanishing  point  ? 

18.  What  is  an  elevation  ? 

19.  What  is  a  plan  ?  What  must  be  indicated  in  the  plan  of  an  object  ?  At  how 
many  points  was  the  building  represented  in  Fig.  3  cut  to  enable  us  to  construct  the 
plan  represented  in  Fig.  5  ?    At  what  points  was  the  building  supposed  to  be  cut  ? 

Turn  to  Figs.  2,  4,  and  5.  In  what  is  the  length  of  the  building  shown  ?  In  what 
is  the  breadth  of  the  building  shown  ?  In  what  is  the  height  of  the  walls  shown  ? 
In  what  is  the  vertical  height  of  the  roof  shown  ?  In  what  is  the  pitch  of  the  roof 
shown  ?  In  what  is  the  height  of  the  chimney  shown  ?  In  what  is  the  width  of  the 
front  and  back  face  of  the  chimney  shown  ?  In  what  is  the  width  of  the  faces  of  the 
chimney  parallel  to  the  end  walls  shown  ?  In  what  is  the  width  of  the  windows 
shown  ?  In  what  is  the  width  of  the  door  shown  ?  In  what  is  the  situation  of  the 
windows  and  door  shown  ?  In  what  are  the  height  of  the  door,  the  height  of  the  win- 
dows, and  their  height  above  the  ground  shown  ? 

CHAPTER  II. 

Of  what  does  this  chapter  treat  ? 

20.  Upon  what  science  is  the  art  of  linear  perspective  based  ?  State  the  facts 
connected  with  vision. 

21.  Draw  upon  the  board  a  diagram  explaining  the  phenomenon  of  vision. 

22.  What  is  said  in  regard  to  viewing  an  object  through  a  transparent  plane  ?  Go 
to  the  board  and  illustrate  this  truth.  What  general  truth  is  deduced  from  this  inves- 
tigation ? 

23.  What  is  said  in  regard  to  finding  the  perspectives  of  lines,  straight  or  curved  ? 

24.  What  part  of  Fig.  8  represents  the  square  which  it  is  proposed  to  place  in 
perspective  ?  How  is  the  square  supposed  to  be  situated  ?  What  is  supposed  to  rep- 
resent the  ground  plane  ?  What  part  of  the  figure  represents  the  picture  plane  ? 
What  relation  is  the  plane  M  B  L  N  supposed  to  bear  to  the  plane  of  the  paper  upon 
which  the  square  is  traced  ?  Where  is  the  observer  supposed  to  stand  ?  Is  the  point 
A  supposed  to  be  on  the  same  plane  as  the  square  ?  At  what  height  is  the  eye  sup- 
posed to  be  above  the  point  A  ?  What  relation  would  a  line  drawn  from  the  eye  to 
the  point  C  on  the  picture  plane  bear  to  that  plane  ?  What  is  the  point  C  called  ? 
What  line  in  this  figure  measures  the  distance  of  picture  ?  What  is  the  figure  F  e  ^  D 
called  ?  What  are  the  lines  F  C  and  D  C  called  ?  Show  how  the  perspective  of  the 
square  is  determined. 

25.  What  relation  do  the  sides  E  F  and  I  D  of  the  square  bear  to  the  picture 
plane  ?    To  what  point  do  their  perspectives  tend  ?    Is  the  truth  here  recognized  a 


QUESTIONS. 


Gl 


general  principle,  or  does  it  apply  simply  to  the  perspectives  of  the  lines  E  F  and 
I  D?  State  the  rule  expressing  this  general  truth.  What  relation  do  the  sides  E  1 
and  F  D  of  the  square  bear  to  the  picture  plane  ?  What  relation  do  the  perspectives 
of  these  lines  bear  to  the  lines  themselves  ?  Is  the  fact  here  noticed  a  general  truth  ? 
State  the  rule  expressing  the  principle  here  recognized. 

26.  What  is  the  distinction  between  parallel  and  oblique  perspective  ?  What  part 
of  Fig.  11  represents  the  squiu-e  which  it  is  projjosed  to  place  in  perspective?  From 
what  point  is  this  figure  supposed  to  be  seen  ?  Since  the  points  S,  U,  O,  and  T  are 
in  the  centres  of  the  sides  of  the  square  E  F  D  I,  where  will  their  perspectives  be 
found  ?    Show  how  the  perspective  of  the  square  is  determined. 

27.  What  relation  do  the  sides  of  the  square  bear  to  the  picture  plane?  On  pro 
longing  the  perspectives  of  the  sides  of  the  square,  in  the  direction  of  tlieir  conver- 
gence, where  do  they  vanish  ?  What  is  the  distance  of  their  vanishing  points  from 
the  centre  of  view,  compared  with  the  distance  of  picture  ?  How  may  this  be  shown  ? 
Is  the  truth  here  recognized  a  general  principle,  or  is  it  limited  in  its  application  to 
the  perspectives  of  the  sides  of  the  square  ?  State  the  rule  expressing  this  law  of 
perspective.  Name  the  sides  of  the  square  which  tend  to  the  right  as  they  recede. 
On  which  side  of  the  centre  of  view  is  the  vanishing  point  of  their  perspectives  ? 
Name  the  sides  of  the  square  which  tend  to  the  left  as  they  recede.  On  which  side  of 
the  centre  of  view  is  the  vanishing  point  of  their  perspectives  ?  Is  the  truth  here 
recognized  a  general  truth  ?    State  the  rule  expressing  this  principle. 

28.  State  Rule  5.    What  are  the  facts  upon  which  this  rule  is  based  ? 

CHAPTER  III. 
Of  what  does  this  chapter  treat  ? 

29.  In  preparing  the  paper,  what  line  is  first  drawn?  In  what  position,  and 
where  on  the  paper,  is  this  line  to  be  drawn  ?  What  does  that  part  of  the  pai)er  above 
this  line  represent  ?  What  does  that  part  of  the  paper  below  this  line  represent  ? 
What  relation  are  these  planes  supposed  to  bear  to  each  other  ?  Of  two  points  placed 
at  unequal  distances  below  the  base  line,  which  point  is  nearer  to  the  picture  plane  ? 
Where  is  the  point  A  in  Fig.  12  ?  Is  the  point  F  to  be  understood  as  bemg  in  front, 
or  behind  the  picture  plane  ? 

30.  Having  drawn  a  line  to  represent  the  base  line  of  the  picture  plane,  what  is 
the  next  step  in  preparing  the  paper  ?  What  determines  the  location  of  the  point 
which  represents  the  centre  of  view  ?  Why  does  the  place  of  the  eye  determine  tliis 
point?  What  determines  the  place  of  the  horizon  ?  For  what  reason?  In  Fig.  12, 
where  is  the  place  of  the  eye  supposed  to  be  ?  What  does  the  line  H  R  in  this  figure 
represent?  What  point  in  Fig.  11  corresponds  to  the  point  C  in  Fig.  12?  Wliat 
relation  must  a  line  bear  to  the  picture  plane  to  have  its  perspective  tend  to  the  point 
C,  in  Fig.  12? 

31.  What  is  the  next  step  in  the  preparation  of  the  paper?  Suppose  it  is 
required  to  find  the  perspective  of  a  cube  seen  on  the  right  of  the  eye,  and  so  situated 
that  one  of  its  faces  makes  with  the  picture  plane  an  angle  of  twenty  degrees,  and 


62 


LINEAR  PERSPECTIVE. 


the  point  in  the  cube  nearest  the  plane  is  distant  from  the  plane  the  measure  of  one 
of  the  edges  of  the  cube  ;  how  should  the  plan  be  drawn  ?  Draw  upon  the  board  a 
plan  of  a  cube  thus  seen,  making  the  angle  as  near  as  you  can  judge  by  the  eye,  and 
let  the  distance  of  the  cube  on  the  right  of  the  eye  be  more  or  less,  as  you  please. 
When  elevations  are  required,  where  are  they  to  be  placed  ? 

32.  Having  drawn  a  plan  of  the  object  to  be  placed  in  perspective,  what  is  the 
next  step  in  the  process?  What  is  the  assumed  distance  of  picture  in  Fig.  12? 
Give  the  exact  place  of  the  eye.  What  is  the  point  D,  which  marks  the  distance  of 
picture,  called?  To  what  point  in  Fig.  11  does  the  point  D  correspond?  What 
must  be  the  direction  of  a  line,  what  must  be  its  position,  and  what  relation  must  it 
bear  to  the  picture  plane,  to  have  its  perspective  tend  to  the  point  D,  in  Fig.  12  ? 

33.  Have  we  alluded  to  all  the  lines  and  points  required  in  preparing  the  paper  ? 
Name  the  lines  and  points. 

CHAPTER  IV. 

Of  what  does  this  chapter  treat  ? 

34.  What  is  the  form  of  the  picture  made  on  the  retina  of  the  eye  ?  What  point 
in  the  scene  is  pictured  in  the  centre  of  the  image  formed  in  the  eye  ?  What  con- 
clusion is  drawn  from  these  facts  in  regard  to  assuming  a  point  as  the  centre  of  view? 

35.  What  remarks  are  made  touching  the  application  of  this  rule  ? 

36.  Upon  what  does  the  truthfulness  of  a  drawing  depend?  What  remark  is 
made  on  the  distance  often  assumed  in  pictures  ?  How  much  of  a  scene  can  be  rep- 
resented in  one  picture  with  accuracy  ?  In  order  to  obtain  a  clear  view  of  all  parts 
of  an  object  within  the  range  of  vision,  how  far  must  the  object  be  from  the  eye  ? 
Have  you  tried  the  experiment  ?  What  is  Fig.  13  intended  to  illustrate  ?  Draw  this 
figure  upon  the  board,  and  explain.  What  is  the  ri^le  for  assuming  the  distance  of 
picture  ? 

CHAPTER  V. 

Of  what  does  this  chapter  treat  ? 

37.  What  is  the  principle  on  which  the  perspectives  of  points  are  found  ?  What 
follows  from  this  principle  ? 

38.  In  the  problems  throughout  the  book,  certain  lines  and  points  are  to  be  under- 
stood as  given,  as  shown  in  the  diagrams  to  which  the  problems  refer ;  what  are  these 
lines  and  points,  and  by  what  letters  are  they  in  all  cases  to  be  distinguished  ? 

39.  State  Prob.  1.  Go  to  the  board  and  work  out  this  problem,  explaining  each 
step  in  the  process. 

40.  What  is  said  respecting  the  line  J  A,  in  Fig.  15  ?  Why  is  it  used  ?  In  de- 
termining the  perspective  of  points,  are  you  to  draw  this  line  ?  What  course  are  you 
to  take? 

41.  State  Prob.  2.  Go  to  the  board  and  work  out  this  problem,  explaining  each 
step  in  the  process.  In  determining  the  perspective  of  the  square,  you  first  found  the 
perspective  of  the  point  A ;  why  not  begin  with  the  point  E  or  I  ?  Is  the  drawing 
rendered  more  simple  by  the  course  pursued  ? 


QUESTIONS. 


G3 


42.  State  Prob.  3.  Go  to  llio  boai-il  and  work  out  tliis  proljluin,  explaining  each 
step  in  the  process.  Instead  of  first  finding  tlie  perspectives  of  the  points  A  and  E, 
would  it  have  been  just  as  well  to  have  taken  the  points  I  and  F?  In  Fig.  IG,  the 
perspective  of  the  point  F  is  found  in  the  line  M  D ;  why  is  not  the  perspective  of 
the  point  F,  in  Fig.  17,  found  in  the  line  M  D  ? 

43.  State  Prob.  4.  Go  to  the  board  and  work  out  this  problem,  explaining  each 
step  in  the  process. 

44.  What  is  the  line  o  a,  in  Fig.  18,  the  perspective  of?  Supjiose  it  were  required 
to  find  the  perspective  of  a  vertical  line  standing  on  a  given  point  on  the  ground 
plane ;  how  Would  you  proceed  ? 

45.  State  Prob.  5.  Are  elevations  required  in  placing  a  cube  in  perspective? 
Why  are  they  not  required  ?  Go  to  the  board  and  work  out  this  problem,  explaining 
each  step  in  the  process. 

46.  Point  out  the  perspectives  of  those  faces  of  the  cube  which  present  a  front 
view ;  what  is  their  form  compared  with  the  form  of  the  faces  represented  ?  What  is 
evident  from  this  ? 

47.  Point  out  the  perspectives  of  the  edges  which  present  a  front  view  ?  What 
is  their  position  compared  with  the  position  of  the  edges  represented  ?  What  is  evi- 
dent from  this  ? 

48.  How  do  the  perspectives  of  the  edges  nearest  the  eye  compare  with  those  of 
the  edges  most  distant?    What  is  evident  from  this? 

49.  How  are  the  left,  right,  upper  and  lower  faces  of  the  cube  seen  ?  Which  face 
is  seen  the  most  obliquely  ?  How  are  these  faces  affected  by  being  thus  seen  ?  Wliat 
is  meant  by  the  term  foreshortened  ?  Which  face  is  the  most  foreshortened  ?  What 
is  evident  from  this  ? 

50.  Is  the  upper  face  of  the  cube  above  or  below  the  level  of  the  eye  ?  Do  the 
perspectives  of  the  receding  sideg  of  the  upper  face  incline  up  or  down  ?  Is  the  lower 
face  of  the  cube  above  or  below  the  level  of  the  eye  ?  Do  the  perspectives  of  the 
receding  sides  of  this  face  incline  up  or  down  ?  What  is  the  conclusion  drawn  front) 
these  facts  ? 

51.  In  what  position  do  all  horizontal  lines  on  a  level  with  the  eye  appear?  From 
what  is  this  evident  ? 

52.  In  sketching,  upon  what  does  the  draughtsman  rely  in  determining  the  per- 
spectives of  objects  ?  Does  one  readily  recognize  the  forms  which  objects  present  to 
the  eye,  without  discipline  ?  Why  should  one  obtain  a  knowledge  of  the  principles 
of  perspective  before  he  attempts  to  draw  from  objects  ? 

53.  You  are  advised  to  draw  from  a  cube  seen  as  that  represented  in  Fig.  20. 
How  is  the  cube  seen  ?  In  drawing  from  an  object,  what  is  the  first  thing  to  be 
considered?  What  part  of  an  object  should  be  first  represented?  Draw  upon  the 
board  the  outlmes  of  a  cube  seen  as  in  Fig.  20,  and  give  your  reasons  for  each  step 
in  the  process. 

54.  How  may  the  accuracy  of  a  drawing  be  determined  ? 

55.  In  sketching,  what  objects  are  recommended  as  suitable  subjects  for  first  prac- 
tice ?  What  directions  are  given  in  regard  to  the  method  of  drawing  objects  ?  Pomt 
out  the  course  pursued  in  drawing  the  chest  represented  in  Fig.  21  ?    How  is  the 


64 


LINEAR  PERSPECTIVE. 


inclination  of  a  line  most  readily  and  accurately  determined  ?  In  case  the  line  to  be 
represented  is  not  connected  with  either  a  horizontal  or  a  vertical  line,  what  is  to  be 
done  ?    What  does  the  method  of  drawing  the  chest  suggest  ? 

56.  What  is  said  in  regard  to  Fig.  22  ?  What  is  one  of  the  objects  of  this  kind 
of  practice  ? 

57.  State  Prob.  6.  Go  to  the  board  and  work  out  this  problem,  explaining  each 
step  in  the  process. 

58.  When  an  error  is  made  in  the  drawing  of  a  pyramid,  in  what  does  it  usually 
consist?    What  is  the  object  of  Fig.  24  ?    Explain  the  method  of  drawing  the  roof? 

59.  State  Prob.  7.  Point  out  the  course  pursued  in  determining  the  perspective 
of  the  building,  referring  to  Fig.  25,  giving  the  reasons  for  every  step  in  the  process. 

60.  State  Prob.  8.  Explain  the  plan  of  the  building  in  Fig.  26.  Is  the  point  of 
distance  represented  in  this  figure  ?  Where  is  this  point  situated  ?  How  may  this 
point  be  determined  ?  Point  out  the  course  pursued  in  determining  the  perspective 
of  the  building  referring  to  Fig.  26,  giving  the  reasons  for  every  step  in  the  process. 

61.  State  Prob.  9.  Explain  the  plan  of  the  building  in  Fig.  27.  Point  out  the 
course  pursued  in  determining  the  perspective  of  the  building,  giving  the  reasons  for 
every  step  in  the  process. 

62.  In  sketching  a  building,  how  far  should  the  draughtsman  be  removed  from  it  ? 
Go  to  the  board  and  point  out  the  order  in  which  the  general  outlines  of  a  building 
should  be  drawn,  when  the  building  is  so  situated  that  the  end  wall  presents  a  front 
view ;  when  so  situated  that  the  front  wall  presents  a  front  view. 

63.  State  Prob.  10.  Is  this  cube  seen  in  parallel  or  in  oblique  perspective?  Go 
to  the  board  and  work  out  this  problem,  explaining  each  step  in  the  process. 

64.  How  is  the  cube  represented  in  Fig.  31  seen?  Draw  upon  the  boarcl  the  out- 
lines of  a  cube  thus  seen,  explaining  each  step  in  the  process. 

65.  In  drawing  a  chair  such  as  we  have  represented  in  Fig.  32,  how  do  you  pro- 
ceed ?    What  course  would  you  pursue  in  drawing  a  *table  or  stand  ? 

66.  What  is  said  in  regard  to  taking  Fig.  32  as  an  exercise  in  drawing  from  the 
copy? 

67.  State  Prob.  11.  Point  out  the  course  pursued  in  determining  the  perspective 
of  the  building,  giving  the  reasons  for  each  step  in  the  process. 

68.  In  sketching  a  building  of  the  usual  form,  in  what  order  are  you  instructed  to 
draw  the  lines  ?    Explain  the  use  of  the  dotted  lines  in  Figs.  35  and  36. 

69.  State  Prob.  12.  How  is  the  perspective  of  a  circle  determined?  How  many 
points  are  usually  taken  in  the  circumference  of  the  circle  ?  How  are  these  points 
determined?  Point  out  the  course  pursued  in  determining  the  perspective  of  the 
circle,  giving  the  reasons  for  each  step  in  the  process. 

70.  How  may  the  perspective  of  an  octagon  be  determined  ?  i 

71.  In  the  examination  of  the  perspective  of  the  circle,  to  what  points  is  attention 
called  ?  i 

72.  In  sketching,  how  may  the  perspective  of  the  circle  be  determined  ? 

73.  State  Prob.  13.  Point  out  the  course  pursued  in  determining  the  perspective 
of  the  cylinder,  giving  the  reasons  for  each  step  in  the  process. 

74.  In  sketching  an  object,  such  as  we  have  represented  in  Fig.  40,  how  are  you 
instructed  to  proceed  ? 


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